Search: a010553 -id:a010553
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A335831
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Numbers k with a record value of tau(tau(k)) (A010553), where tau(k) is the number of divisors of k (A000005).
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+20
2
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1, 2, 6, 12, 60, 360, 1260, 2520, 5040, 55440, 277200, 720720, 3603600, 61261200, 129729600, 908107200, 2205403200, 15437822400, 293318625600, 3226504881600, 6746328388800, 74209612276800, 195643523275200, 1855240306920000, 2152078756027200, 27977023828353600
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OFFSET
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1,2
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COMMENTS
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First differs from A189394 at n=15.
The corresponding record values are 1, 2, 3, 4, 6, 8, 9, 10, 12, 16, 18, 20, 24, 30, ... (see the link for more values).
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LINKS
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Yvonne Buttkewitz, Christian Elsholtz, Kevin Ford and Jan-Christoph Schlage-Puchta, A problem of Ramanujan, Erdős, and Kátai on the iterated divisor function, International Mathematics Research Notices, Vol. 2012, No. 17 (2012), pp. 4051-4061, preprint, arXiv:1108.1815 [math.NT], 2011.
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FORMULA
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tau(tau(a(n))) ~ c * sqrt(log(a(n)))/log(log(a(n))), where c is a constant (Buttkewitz et al., 2012).
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MATHEMATICA
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f[n_] := DivisorSigma[0, DivisorSigma[0, n]]; fm = 0; s = {}; Do[f1 = f[n]; If[f1 > fm, fm = f1; AppendTo[s, n]], {n, 1, 10^5}]; s
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A083399
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Number of divisors of n that are not divisors of other divisors of n.
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+10
19
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1, 2, 2, 2, 2, 3, 2, 2, 2, 3, 2, 3, 2, 3, 3, 2, 2, 3, 2, 3, 3, 3, 2, 3, 2, 3, 2, 3, 2, 4, 2, 2, 3, 3, 3, 3, 2, 3, 3, 3, 2, 4, 2, 3, 3, 3, 2, 3, 2, 3, 3, 3, 2, 3, 3, 3, 3, 3, 2, 4, 2, 3, 3, 2, 3, 4, 2, 3, 3, 4, 2, 3, 2, 3, 3, 3, 3, 4, 2, 3, 2, 3, 2, 4, 3, 3, 3, 3, 2, 4, 3, 3, 3, 3, 3, 3, 2, 3, 3, 3, 2, 4
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OFFSET
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1,2
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COMMENTS
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a(n)<=tau(n); a(n)=tau(n) iff n is prime or n=1 (A008578, A000040); a(n)=tau(n)-1 iff n is semiprime (A001358).
a(n) is the number of maximal subsemigroups of the annular Jones monoid of degree n.
a(n) is the number of maximal subsemigroups of the monoid of orientation-preserving mappings on a set with n elements.
a(n) + 1 is the number of maximal subsemigroups of the monoid of orientation-preserving partial mappings on a set with n elements.
(End)
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LINKS
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FORMULA
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a(n) = omega(n) + 1, where omega = A001221.
G.f.: x/(1 - x) + Sum_{k>=1} x^prime(k)/(1 - x^prime(k)). - Ilya Gutkovskiy, Mar 21 2017
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EXAMPLE
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{1,2,3,4,6,8,12,24} are the divisors of n=24: 1, 2, 3, 4 and 6 divide not only 24, but also 8 or 12, therefore a(24)=3.
{1,2,3,4,6,8,12,24} are the divisors of n=24: 1, 2 and 3 are noncomposites, therefore a(24)=3. - Jaroslav Krizek, Nov 25 2009
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MAPLE
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1+nops(numtheory[factorset](n)) ;
end proc:
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MATHEMATICA
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Rest@ CoefficientList[ Series[(1/(1 - x)) + Sum[1/(1 - x^Prime[j]), {j, 200}], {x, 0, 111}], x] (* Robert G. Wilson v, Aug 16 2011 *)
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PROG
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(Haskell)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A212171
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Prime signature of n (nonincreasing version): row n of table lists positive exponents in canonical prime factorization of n, in nonincreasing order.
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+10
17
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1, 1, 2, 1, 1, 1, 1, 3, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 4, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 3, 1, 2, 1, 1, 3, 2, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 4, 1, 2, 2, 1, 1, 1, 2, 1, 1, 3, 1, 1, 1, 3, 1
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OFFSET
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2,3
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COMMENTS
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The multiset of positive exponents in n's prime factorization completely determines a(n) for a host of OEIS sequences, including several "core" sequences. Of those not cross-referenced here or in A212172, many can be found by searching the database for A025487.
(Note: Differing opinions may exist about whether the prime signature of n should be defined as this multiset itself, or as a symbol or collection of symbols that identify or "signify" this multiset. The definition of this sequence is designed to be compatible with either view, as are the original comments. When n >= 2, the customary ways to signify the multiset of exponents in n's prime factorization are to list the constituent exponents in either nonincreasing or nondecreasing order; this table gives the nonincreasing version.)
Table lists exponents in the order in which they appear in the prime factorization of a member of A025487. This ordering is common in database comments (e.g., A008966).
Each possible multiset of an integer's positive prime factorization exponents corresponds to a unique partition that contains the same elements (cf. A000041). This includes the multiset of 1's positive exponents, { } (the empty multiset), which corresponds to the partition of 0.
Differs from A124010 from a(23) on, corresponding to the factorization of 18 = 2^1*3^2 which is here listed as row 18 = [2, 1], but as [1, 2] (in the order of the prime factors) in A124010 and also in A118914 which lists the prime signatures in nondecreasing order (so that row 12 = 2^2*3^1 is also [1, 2]). - M. F. Hasler, Apr 08 2022
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LINKS
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FORMULA
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Row n of A124010 for n > 1, with exponents sorted in nonincreasing order. Equivalently, row A046523(n) of A124010 for n > 1.
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EXAMPLE
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First rows of table read:
1;
1;
2;
1;
1,1;
1;
3;
2;
1,1;
1;
2,1;
...
The multiset of positive exponents in the prime factorization of 6 = 2*3 is {1,1} (1s are often left implicit as exponents). The prime signature of 6 is therefore {1,1}.
12 = 2^2*3 has positive exponents 2 and 1 in its prime factorization, as does 18 = 2*3^2. Rows 12 and 18 of the table both read {2,1}.
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PROG
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(Magma) &cat[Reverse(Sort([pe[2]:pe in Factorisation(n)])):n in[1..76]]; // Jason Kimberley, Jun 13 2012
(PARI) apply( {A212171_row(n)=vecsort(factor(n)[, 2]~, , 4)}, [1..40])\\ M. F. Hasler, Apr 19 2022
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CROSSREFS
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A212172 cross-references over 20 sequences that depend solely on n's prime exponents >= 2, including the "core" sequence A000688. Other sequences determined by the exponents in the prime factorization of n include:
Other: A001055, A008480, A010553, A038548, A050320, A051707, A071625, A074206, A076078, A085082, A088873.
A highly incomplete selection of sequences, each definable by the set of prime signatures possessed by its members: A000040, A000290, A000578, A000583, A000961, A001248, A001358, A001597, A001694, A002808, A004709, A005117, A006881, A013929, A030059, A030229, A052486.
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KEYWORD
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nonn,easy,tabf
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AUTHOR
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STATUS
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approved
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A036450
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a(n) = d(d(d(n))), the 3rd iterate of the number-of-divisors function with an initial value of n.
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+10
15
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1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 3, 2, 3, 2, 2, 2, 3, 2, 2, 2, 3, 2, 3, 2, 3, 2, 2, 2, 2, 2, 2, 2, 3, 2, 3, 2, 3, 3, 2, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 2, 2, 4, 2, 2, 3, 2, 2, 3, 2, 3, 2, 3, 2, 4, 2, 2, 3, 3, 2, 3, 2, 3, 2, 2, 2, 4, 2, 2, 2, 3, 2, 4, 2, 3, 2, 2, 2, 4, 2, 3, 3, 2, 2, 3, 2, 3, 3
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OFFSET
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1,2
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COMMENTS
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The iterated d function rapidly converges to the fixed point 2.
The fourth iterate begins as follows:
1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, ... . (End)
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REFERENCES
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S. Ramanujan, Collected Papers, Ed. G. H. Hardy et al., Cambridge 1927; Chelsea, NY, 1962, p. 128. - N. J. A. Sloane, Jun 02 2014
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LINKS
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EXAMPLE
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n = 5040, d(5040) = 60, d(d(5040)) = d(60) = 12 and a(5040) = d(12) = 6.
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MATHEMATICA
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PROG
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(Python)
from sympy import divisor_count
def A036450(n): return divisor_count(divisor_count(divisor_count(n))) # Chai Wah Wu, Nov 17 2022
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A036452
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a(n) = d(d(d(d(n)))), the 4th iterate of number-of-divisors function with initial value of n.
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+10
11
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1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2
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OFFSET
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1,2
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COMMENTS
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The iterated d function rapidly converges to fixed point 2. For k=4, the first n for which a(n)>2 is 60.
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LINKS
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FORMULA
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a(n) = d(d(d(d(n)))).
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EXAMPLE
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E.g., n=96 and its successive iterates are 12,6,4,3 and 2. The 4th term is a(96)=3.
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MATHEMATICA
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PROG
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A036453
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a(n) = d(d(d(d(d(n))))), the 5th iterate of the number-of-divisors function d = A000005, with initial value n.
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+10
9
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1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2
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OFFSET
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1,2
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COMMENTS
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The iterated d function rapidly converges to fixed point 2. In the 5th iterated d-sequence, the first term different from the fixed point 2 appears at n = 5040. The 6th and further iterated sequences have very long initial segment of 2's. In the 6th one the first non-stationary term is a(293318625600) = 3. In such sequences any large value occurs infinite many times and constructible.
Differs from A007395 for n = 1, 5040, 7920, 8400, 9360, 10080, 10800, etc. - R. J. Mathar, Oct 20 2008
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LINKS
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EXAMPLE
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E.g., n = 96 and its successive iterates are 12, 6, 4, 3 and 2. The 5th term is a(96) = 2 is stationary (fixed).
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MATHEMATICA
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Table[Nest[DivisorSigma[0, #]&, n, 5], {n, 110}] (* Harvey P. Dale, Jun 18 2021 *)
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PROG
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A036454
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Prime powers with special exponents: q^(p-1) where p > 2 and q are prime numbers.
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+10
7
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4, 9, 16, 25, 49, 64, 81, 121, 169, 289, 361, 529, 625, 729, 841, 961, 1024, 1369, 1681, 1849, 2209, 2401, 2809, 3481, 3721, 4096, 4489, 5041, 5329, 6241, 6889, 7921, 9409, 10201, 10609, 11449, 11881, 12769, 14641, 15625, 16129, 17161, 18769, 19321
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OFFSET
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1,1
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COMMENTS
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Composite numbers with a prime number of divisors.
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LINKS
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FORMULA
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d(d(a(n))) = 2, where d(x) = tau(x) = sigma_0(x) is the number of divisors of x.
Sum_{n>=1} 1/a(n) = Sum_{k>=2} P(prime(k)-1) = 0.54756961912815344341..., where P is the prime zeta function. - Amiram Eldar, Jul 10 2022
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EXAMPLE
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From powers of 2: 4,16,64,1024,4096,65536 are in the sequence since exponent+1 is also prime. The same powers of any prime base are also included.
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MAPLE
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N:= 10^5:
P1:= select(isprime, [2, seq(2*i+1, i=1..floor((sqrt(N)-1)/2))]):
P2:= select(`<=`, P1, 1+ilog2(N))[2..-1]:
S:= {seq(seq(p^(q-1), q = select(`<=`, P2, 1+floor(log[p](N)))), p=P1)}:
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MATHEMATICA
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specialPrimePowerQ[n_] := With[{f = FactorInteger[n]}, Length[f] == 1 && PrimeQ[f[[1, 1]]] && f[[1, 2]] > 1 && PrimeQ[f[[1, 2]] + 1]]; Select[Range[20000], specialPrimePowerQ] (* Jean-François Alcover, Jul 02 2013 *)
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PROG
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(PARI) for(n=1, 34000, if(isprime(n), n++, x=numdiv(n); if(isprime(x), print(n))))
(PARI) list(lim)=my(v=List(), t); lim=lim\1+.5; forprime(p=3, log(lim)\log(2) +1, t=p-1; forprime(q=2, lim^(1/t), listput(v, q^t))); vecsort(Vec(v))
(Haskell)
a009087 n = a009087_list !! (n-1)
a009087_list = filter ((== 1) . a010051 . (+ 1) . a100995) a000961_list
(Magma) [n: n in [1..20000] | not IsPrime(n) and IsPrime(DivisorSigma(0, n))]; // Vincenzo Librandi, May 19 2015
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CROSSREFS
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KEYWORD
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nonn,easy,nice
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AUTHOR
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STATUS
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approved
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A141113
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Positive integers k such that d(d(k)) divides k, where d(k) is the number of divisors of k.
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+10
4
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1, 2, 4, 6, 12, 15, 16, 20, 21, 24, 27, 28, 32, 33, 36, 39, 40, 44, 48, 51, 52, 56, 57, 60, 64, 68, 69, 72, 76, 80, 84, 87, 88, 90, 92, 93, 96, 104, 108, 111, 112, 116, 120, 123, 124, 126, 128, 129, 132, 136, 141, 144, 148, 150, 152, 156, 159, 164, 172, 176, 177, 180
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OFFSET
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1,2
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LINKS
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EXAMPLE
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28 has 6 divisors and 6 has 4 divisors. 4 divides 28, so 28 is in the sequence.
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MAPLE
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with(numtheory): a:=proc(n) if `mod`(n, tau(tau(n))) = 0 then n else end if end proc: seq(a(n), n=1..200); # Emeric Deutsch, Jun 05 2008
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MATHEMATICA
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Select[Range[200], Divisible[#, DivisorSigma[0, DivisorSigma[0, #]]]&] (* Harvey P. Dale, Feb 05 2012 *)
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PROG
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(PARI) is(k) = k%numdiv(numdiv(k)) == 0; \\ Jinyuan Wang, Feb 19 2019
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A141114
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Positive integers k where d(d(k)) is coprime to k, where d(k) is the number of divisors of k.
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+10
3
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1, 3, 5, 7, 8, 9, 10, 11, 13, 14, 17, 19, 22, 23, 25, 26, 29, 31, 34, 35, 37, 38, 41, 43, 45, 46, 47, 49, 53, 55, 58, 59, 61, 62, 63, 65, 67, 71, 73, 74, 75, 77, 79, 81, 82, 83, 85, 86, 89, 91, 94, 95, 97, 99, 100, 101, 103, 105, 106, 107, 109, 113, 115, 117, 118, 119, 121
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OFFSET
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1,2
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COMMENTS
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Includes all primes, squares of odd primes, and squarefree semiprimes coprime to 3. - Robert Israel, Dec 16 2019
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LINKS
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EXAMPLE
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26 has 4 divisors and 4 has 3 divisors. 3 is coprime to 26, so 26 is in the sequence.
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MAPLE
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filter:= proc(n) uses numtheory;
igcd(tau(tau(n)), n) = 1
end proc:
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MATHEMATICA
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Select[Range[200], GCD[DivisorSigma[0, DivisorSigma[0, # ]], # ] == 1 &] (* Stefan Steinerberger, Jun 05 2008 *)
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PROG
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(Magma) [k:k in [1..130]|Gcd(k, #Divisors(#Divisors(k))) eq 1]; // Marius A. Burtea, Dec 16 2019
(PARI) is(n) = gcd(numdiv(numdiv(n)), n)==1 \\ Felix Fröhlich, Dec 16 2019
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A141115
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Those positive integers k where both d(d(k)) is not coprime to k and d(d(k)) does not divide k, where d(k) is the number of divisors of k.
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+10
3
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18, 30, 42, 50, 54, 66, 70, 78, 98, 102, 110, 114, 130, 138, 140, 154, 160, 162, 170, 174, 182, 186, 190, 200, 220, 222, 224, 230, 238, 242, 246, 250, 258, 260, 266, 282, 286, 290, 308, 310, 315, 318, 322, 338, 340, 350, 352, 354, 364, 366, 370, 374, 380, 392
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OFFSET
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1,1
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LINKS
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EXAMPLE
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50 has 6 divisors and 6 has 4 divisors. 4 is not coprime to 50 and 4 does not divide 50. So 50 is in the sequence.
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MATHEMATICA
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Select[Range[400], GCD[DivisorSigma[0, DivisorSigma[0, # ]], # ] > 1 && Mod[ #, DivisorSigma[0, DivisorSigma[0, # ]]] > 0 &] (* Stefan Steinerberger, Jun 05 2008 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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