Displaying 1-10 of 16 results found.
Triangle in which first row is 0, n-th row (n>1) lists the exponents of distinct prime factors ("ordered prime signature") in the prime factorization of n.
+10
451
0, 1, 1, 2, 1, 1, 1, 1, 3, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 4, 1, 1, 2, 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, 1, 2, 1, 1, 2, 1, 1, 1, 3, 1, 1, 3, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 6, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 3, 2, 1
COMMENTS
A001222(n) = Sum(T(n,k), 1 <= k <= A001221(n)); A005361(n) = Product(T(n,k), 1 <= k <= A001221(n)), n>1; A051903(n) = Max(T(n,k): 1 <= k <= A001221(n)); A051904(n) = Min(T(n,k), 1 <= k <= A001221(n)); A067029(n) = T(n,1); A071178(n) = T(n, A001221(n)); A064372(n)=Sum( A064372(T(n,k)), 1 <= k <= A001221(n)). - Reinhard Zumkeller, Aug 27 2011
Any finite sequence of natural numbers appears as consecutive terms. - Paul Tek, Apr 27 2013 Most often the prime signature is given as a sorted representative of the multiset of the nonzero exponents, either in increasing order, which yields A118914, or, most commonly, in decreasing order, which yields A212171. - M. F. Hasler, Oct 12 2018
EXAMPLE
Initial values of exponents are:
1, [0]
2, [1]
3, [1]
4, [2]
5, [1]
6, [1, 1]
7, [1]
8, [3]
9, [2]
10, [1, 1]
11, [1]
12, [2, 1]
13, [1]
14, [1, 1]
15, [1, 1]
16, [4]
17, [1]
18, [1, 2]
19, [1]
20, [2, 1]
...
MAPLE
expts:=proc(n) local t1, t2, t3, t4, i; if n=1 then RETURN([0]); fi; if isprime(n) then RETURN([1]); fi; t1:=ifactor(n); if nops(factorset(n))=1 then RETURN([op(2, t1)]); fi; t2:=nops(t1); t3:=[]; for i from 1 to t2 do t4:=op(i, t1); if nops(t4) = 1 then t3:=[op(t3), 1]; else t3:=[op(t3), op(2, t4)]; fi; od; RETURN(t3); end; # N. J. A. Sloane, Dec 20 2007
MATHEMATICA
row[1] = {0}; row[n_] := FactorInteger[n][[All, 2]] // Flatten; Table[row[n], {n, 1, 80}] // Flatten (* Jean-François Alcover, Aug 19 2013 *)
PROG
(Haskell)
a124010 n k = a124010_tabf !! (n-1) !! (k-1)
a124010_row 1 = [0]
a124010_row n = f n a000040_list where
f 1 _ = []
f u (p:ps) = h u 0 where
h v e | m == 0 = h v' (e + 1)
| m /= 0 = if e > 0 then e : f v ps else f v ps
where (v', m) = divMod v p
a124010_tabf = map a124010_row [1..]
(PARI) print1(0); for(n=2, 50, f=factor(n)[, 2]; for(i=1, #f, print1(", "f[i]))) \\ Charles R Greathouse IV, Nov 07 2014 (Python)
from sympy import factorint
def a(n):
f=factorint(n)
return [0] if n==1 else [f[i] for i in f]
Table of the prime signatures (sorted lists of exponents of distinct prime factors) of the positive integers.
+10
240
1, 1, 2, 1, 1, 1, 1, 3, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 4, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 3, 2, 1, 1, 3, 1, 2, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 4, 2, 1, 2, 1, 1, 1, 2, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 6, 1, 1
COMMENTS
Since the prime factorization of 1 is the empty product (i.e., the multiplicative identity, 1), it follows that the prime signature of 1 is the empty multiset { }. (Cf. http://oeis.org/wiki/Prime_signature)
MathWorld wrongly defines the prime signature of 1 as {1}, which is actually the prime signature of primes.
The sequences A025487, A036035, A046523 consider the prime signatures of 1 and 2 to be distinct, implying { } for 1 and {1} for 2. Since the prime signature of n is a partition of Omega(n), also true for Omega(1) = 0, the order of exponents is only a matter of convention (using reverse sorted lists of exponents would create a different sequence).
Here the multisets of nonzero exponents are sorted in increasing order; it is slightly more common to order them, as the parts of partitions, in decreasing order. This yields A212171. - M. F. Hasler, Oct 12 2018
EXAMPLE
The table starts:
n : prime signature of n (factorization of n)
1 : {}, (empty product)
2 : {1}, (2^1)
3 : {1}, (3^1)
4 : {2}, (2^2)
5 : {1}, (5^1)
6 : {1, 1}, (2^1 * 3^1)
7 : {1}, (5^1)
8 : {3}, (2^3)
9 : {2}, (3^2)
10 : {1, 1}, (2^1 * 5^1)
11 : {1}, (11^1)
12 : {1, 2}, (2^2 * 3^1, but exponents are sorted increasingly)
etc.
MATHEMATICA
primeSignature[n_] := Sort[ FactorInteger[n] , #1[[2]] < #2[[2]]&][[All, 2]]; Flatten[ Table[ primeSignature[n], {n, 2, 65}]](* Jean-François Alcover, Nov 16 2011 *)
PROG
(Haskell)
import Data.List (sort)
a118914 n k = a118914_tabf !! (n-2) !! (k-1)
a118914_row n = a118914_tabf !! (n-2)
a118914_tabf = map sort $ tail a124010_tabf
Row n of table represents second signature of n: list of exponents >= 2 in canonical prime factorization of n, in nonincreasing order, or 0 if no such exponent exists.
+10
26
0, 0, 0, 2, 0, 0, 0, 3, 2, 0, 0, 2, 0, 0, 0, 4, 0, 2, 0, 2, 0, 0, 0, 3, 2, 0, 3, 2, 0, 0, 0, 5, 0, 0, 0, 2, 2, 0, 0, 0, 3, 0, 0, 0, 2, 2, 0, 0, 4, 2, 2, 0, 2, 0, 3, 0, 3, 0, 0, 0, 2, 0, 0, 2, 6, 0, 0, 0, 2, 0, 0, 0, 3, 2, 0, 0, 2, 2, 0, 0, 0, 4, 4, 0, 0, 2, 0, 0
COMMENTS
The multiset of exponents >=2 in the prime factorization of n completely determines a(n) for over 20 sequences in the database (see crossreferences). It also determines the fractions A034444(n)/ A000005(n) and A037445(n)/ A000005(n). For squarefree numbers, this multiset is { } (the empty multiset). The use of 0 in the table to represent each n with no exponents >=2 in its prime factorization accords with the usual OEIS practice of using 0 to represent nonexistent elements when possible. In comments, the second signature of squarefree numbers will be represented as { }.
For each second signature {S}, there exist values of j and k such that, if the second signature of n is {S}, then A085082(n) is congruent to j modulo k. These values are nontrivial unless {S} = { }. Analogous (but not necessarily identical) values of j and k also exist for each second signature with respect to A088873 and A181796. Each sequence of integers with a given second signature {S} has a positive density, unlike the analogous sequences for prime signatures. The highest of these densities is 6/Pi^2 = 0.607927... for A005117 ({S} = { }).
LINKS
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
FORMULA
For nonsquarefree n, row n is identical to row A057521(n) of table A212171.
EXAMPLE
First rows of table read: 0; 0; 0; 2; 0; 0; 0; 3; 2; 0; 0; 2;...
12 = 2^2*3 has positive exponents 2 and 1 in its canonical prime factorization (1s are often left implicit as exponents). Since only exponents that are 2 or greater appear in a number's second signature, 12's second signature is {2}.
30 = 2*3*5 has no exponents greater than 1 in its prime factorization. The multiset of its exponents >= 2 is { } (the empty multiset), represented in the table with a 0.
72 = 2^3*3^2 has positive exponents 3 and 2 in its prime factorization, as does 108 = 2^2*3^3. Rows 72 and 108 both read {3,2}.
MATHEMATICA
row[n_] := Select[ FactorInteger[n][[All, 2]], # >= 2 &] /. {} -> 0 /. {k__} -> Sequence[k]; Table[row[n], {n, 1, 100}] (* Jean-François Alcover, Apr 16 2013 *)
PROG
(Magma) &cat[IsEmpty(e)select [0]else Reverse(Sort(e))where e is[pe[2]:pe in Factorisation(n)|pe[2]gt 1]:n in[1..102]]; // Jason Kimberley, Jun 13 2012
CROSSREFS
Functions determined by exponents >=2 in the prime factorization of n:
Multiplicative: A000688, A005361, A008966, A038538, A046951, A049419, A050361, A050377, A056624, A061704, A063775, A162510, A162511, A212181. Sequences that contain all integers of a specific second signature: A005117 (second signature { }), A060687 ({2}), A048109 ({3}).
Row n of table gives prime metasignature of n: count total appearances of each distinct integer that appears in the prime signature of n, then arrange totals in nonincreasing order.
+10
16
1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 1, 1, 2, 2, 2, 2, 1, 2, 2, 1, 1, 1, 3, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 2, 1, 1, 2, 1, 1, 1, 2, 3, 1, 1, 1, 2, 3, 1, 1
COMMENTS
A prime metasignature is analogous to the signature of a partition (cf. A115621); it is the signature of a prime signature. Row n also gives prime signature of A181819(n).
EXAMPLE
The prime signature of 72 (2^3*3^2) is {3,2}. The numbers 3 and 2 each appear once; therefore, the prime metasignature of 72 is {1,1}.
The prime signature of 120 (2^3*3*5) is {3,1,1}. 3 appears 1 time and 1 appears 2 times; therefore, the prime metasignature of 120 is {2,1}.
CROSSREFS
Length of row n equals A071625(n); sum of numbers in row n is A001221(n).
Row n of table lists exponents in canonical prime factorization of A181800(n) (n-th powerful number that is the first integer of its prime signature), in nonincreasing order.
+10
6
2, 3, 4, 5, 2, 2, 6, 3, 2, 7, 4, 2, 3, 3, 8, 5, 2, 4, 3, 9, 6, 2, 5, 3, 2, 2, 2, 10, 7, 2, 4, 4, 6, 3, 3, 2, 2, 11, 8, 2, 5, 4, 7, 3, 4, 2, 2, 12, 9, 2, 6, 4, 3, 3, 2, 8, 3, 5, 2, 2, 5, 5, 13, 10, 2, 7, 4, 4, 3, 2, 9, 3, 6, 2, 2, 6, 5, 14, 11, 2, 8, 4, 5, 3, 2, 3
EXAMPLE
Since 72 is a member of A181800, all positive exponents in its prime factorization (2^3*3^2) equal or exceed 2. Therefore, its second signature is the same as its prime signature, namely, {3,2} (nonincreasing version). Since 72 = A181800 (8), row 8 represents the prime signature and second signature {3,2}.
a(n) = number of distinct prime signatures represented among divisors of A181800(n) (n-th powerful number that is the first integer of its prime signature).
+10
6
1, 3, 4, 5, 6, 6, 7, 9, 8, 12, 10, 9, 15, 14, 10, 18, 18, 10, 11, 21, 15, 22, 16, 12, 24, 20, 26, 22, 13, 27, 25, 19, 30, 28, 21, 14, 30, 30, 28, 34, 34, 27, 15, 33, 35, 37, 20, 38, 40, 33, 31, 16, 36, 40, 46, 15, 28, 30, 42, 46, 39, 43, 17, 39, 45, 55, 25, 35
COMMENTS
Consider a member of A181800 with second signature {S} whose divisors represent a total of k distinct second signatures and a total of (j+k) distinct prime signatures. Let n be any integer with second signature {S}. Then A212180(n) = k and A085082(n) is congruent to j modulo k. Cf. A212643, A212644.
EXAMPLE
The divisors of 36 represent a total of 6 distinct prime signatures (cf. A085082), as can be seen from the positive exponents, if any, in the canonical prime factorization of each divisor: { }: 1 (multiset of positive exponents is the empty multiset)
{1}: 2 (2^1), 3 (3^1)
{1,1}: 6 (2^1*3^1)
{2}: 4 (2^2), 9 (3^2),
{2,1}: 12 (2^2*3^1), 18 (2^1*3^2)
{2,2}: 36 (2^2*3^2)
Numbers j whose largest divisor <= sqrt(j) is a power of 2.
+10
6
1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 13, 14, 16, 17, 19, 20, 22, 23, 24, 26, 28, 29, 31, 32, 34, 37, 38, 41, 43, 44, 46, 47, 52, 53, 58, 59, 61, 62, 64, 67, 68, 71, 72, 73, 74, 76, 79, 80, 82, 83, 86, 88, 89, 92, 94, 96, 97, 101, 103, 104, 106, 107, 109, 112, 113, 116, 118, 122, 124, 127, 128, 131, 134, 136, 137
COMMENTS
Also indices of the powers of 2 in A033676. Also numbers in increasing order from the columns k of A163280 where k is a power of 2. Observation: at least the first 82 terms of the subsequence of terms with no middle divisors (that is 3, 5, 7, 10, ...) coincide with at least the first 82 terms of A246955. For the definition of middle divisor see A067742. Most of the early terms are in A342081, which consists of powers of 2 together with products of a prime and a power of 2 where the prime is the larger. The exceptions are 24, 72, 80, 96, 112, ... . The odd terms clearly consist of 1 and the odd primes. We can fully characterize the even terms by their A290110 values, which depend on the relative sizes of a number's divisors. A290110 provides a refinement of the classification of numbers by prime signature (cf. A212171): see the example below for numbers with the same prime signature as 48. (End)
EXAMPLE
The table below looks at numbers j with prime signature (4, 1), showing the presence of j and its characterization by A290110(j): 48 = 2^4 * 3 16 no
80 = 2^4 * 5 21 yes
112 = 2^4 * 7 21 yes
162 = 2 * 3^4 36 no
176 = 2^4 * 11 38 no
208 = 2^4 * 13 38 no
272 = 2^4 * 17 51 yes
304 = 2^4 * 19 51 yes
368 = 2^4 * 23 51 yes
...
Clearly any odd composite number is exempted, for example:
891 = 3^4 * 11 21 no
6723 = 3^4 * 83 51 no
Note that A290110(j) = 36 for j = 2 * p^4, prime p; and A290110(j) = 51 for j = 2^4 * p, prime p >= 17. (End)
MATHEMATICA
q[n_] := Module[{d = Divisors[n], mid}, mid = d[[Ceiling[Length[d]/2]]]; mid == 2^IntegerExponent[mid, 2]]; Select[Range[150], q] (* Amiram Eldar, Oct 11 2023 *)
PROG
(PARI) f(n) = local(d); if(n<2, 1, d=divisors(n); d[(length(d)+1)\2]); \\ A033676isp2(n) = 2^logint(n, 2) == n;
(Python)
from itertools import count, islice
from sympy import divisors
def A365406_gen(startvalue=1): # generator of terms >= startvalue return filter(lambda i:(a:=(d:=divisors(i))[len(d)-1>>1])==1<<a.bit_length()-1, count(max(startvalue, 1)))
Irregular triangle read by rows T(n,k): T(1,1) = 0; for n > 1, row n lists exponents of distinct prime factors of the n-th highly composite number ( A002182(n)), where column k = 1, 2, 3, ..., omega( A002182(n)) = A108602(n).
+10
5
0, 1, 2, 1, 1, 2, 1, 3, 1, 2, 2, 4, 1, 2, 1, 1, 3, 1, 1, 2, 2, 1, 4, 1, 1, 3, 2, 1, 4, 2, 1, 3, 1, 1, 1, 2, 2, 1, 1, 4, 1, 1, 1, 3, 2, 1, 1, 4, 2, 1, 1, 3, 3, 1, 1, 5, 2, 1, 1, 4, 3, 1, 1, 6, 2, 1, 1, 4, 2, 2, 1, 3, 2, 1, 1, 1, 4, 4, 1, 1, 5, 2, 2, 1, 4, 2, 1
COMMENTS
For n > 1, row n of table gives the "nonincreasing order" version of the prime signature of A002182(n) (cf. A212171). This order is also the natural order of the exponents in the prime factorization of any highly composite number. The distinct prime factors corresponding to exponents in row n are given by A318490(n, k), where k = 1, 2, 3, ..., A108602(n).
REFERENCES
S. Ramanujan, Highly composite numbers, Proc. Lond. Math. Soc. 14 (1915), 347-409; reprinted in Collected Papers, Ed. G. H. Hardy et al., Cambridge 1927; Chelsea, NY, 1962.
EXAMPLE
First rows read:
0;
1;
2;
1, 1;
2, 1;
3, 1;
2, 2;
4, 1;
2, 1, 1;
3, 1, 1;
2, 2, 1;
4, 1, 1;
...
1st row: A002182(1) = 1 so T(1, 1) = 0; 2nd row: A002182(2) = 2^1 so T(2, 1) = 1; 3rd row: A002182(3) = 4 = 2^2 so T(3, 1) = 2; 4th row: A002182(4) = 6 = 2^1 * 3^1 so T(4, 1) = 1 and T(4, 2) = 1; 5th row: A002182(5) = 12 = 2^2 * 3^1 so T(5, 1) = 2 and T(5, 2) = 1; 6th row: A002182(6) = 24 = 2^3 * 3^1 so T(6, 1) = 3 and T(6, 2) = 1.
CROSSREFS
Row n has length A108602(n), n >= 2.
Number of distinct prime factors of A181800(n) (n-th powerful number that is the first integer of its prime signature).
+10
4
0, 1, 1, 1, 1, 2, 1, 2, 1, 2, 2, 1, 2, 2, 1, 2, 2, 3, 1, 2, 2, 2, 3, 1, 2, 2, 2, 3, 1, 2, 2, 3, 2, 3, 2, 1, 2, 2, 3, 2, 3, 2, 1, 2, 2, 3, 3, 2, 3, 2, 3, 1, 2, 2, 3, 4, 2, 3, 2, 3, 2, 3, 1, 2, 2, 3, 4, 2, 3, 2, 3, 2, 3, 1, 2, 3, 2, 3, 4, 2, 3, 3, 2, 3, 2, 3, 1, 4
COMMENTS
Since each prime factor of A181800(n) divides A181800(n) at least twice, this is also the number of exponents > 2 in prime factorization of A181800(n). Length of row A181800(n) of table A212171 equals a(n) for n > 1. Row A181800(n) of table A212172 has the same length when n > 1 (length = 1 if n = 1).
EXAMPLE
72 (2^3*3^2) has 2 distinct prime factors. Since 72 = A181800(8), a(8) = 2.
Let b(n) and c(n) be the total numbers of distinct prime signatures and second signatures, respectively, represented among divisors of A181800(n) (first integers of each second signature; cf. A212172). b(n) mod c(n) = a(n).
+10
4
0, 1, 1, 1, 1, 0, 1, 4, 1, 5, 4, 1, 6, 5, 1, 7, 6, 2, 1, 8, 5, 7, 2, 1, 9, 6, 8, 2, 1, 10, 7, 1, 9, 2, 6, 1, 11, 8, 0, 10, 2, 7, 1, 12, 9, 18, 0, 11, 2, 8, 15, 1, 13, 10, 22, 0, 7, 14, 12, 2, 9, 20, 1, 14, 11, 26, 7, 8, 18, 13, 2, 10, 25, 1, 15, 15, 12, 30, 9
COMMENTS
Significance of the sequence: Consider a member of A181800 with second signature {S} whose divisors represent a total of k distinct second signatures and a total of (j+k) distinct prime signatures. For all integers n with second signature {S}, A212180(n) = k and A085082(n) is congruent to j modulo k; see examples.
EXAMPLE
4 is the smallest integer with second signature {2}, and its divisors represent 3 distinct prime signatures and 2 distinct second signatures. 1 = 3 mod 2. Since 4 = A181800(2), a(2) = 1. For all integers m with second signature {2}, A085082(m) is congruent to 1 modulo 2. 10800 is the smallest integer with second signature {4,3,2}, and its divisors represent 28 distinct prime signatures and 14 distinct second signatures. 0 = 28 mod 14. Since 10800 = A181800(39), a(39) = 0. For all integers m with second signature {4,3,2}, A085082(m) is congruent to 0 modulo 14.
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