Displaying 1-10 of 61 results found.
Factorial base exp-function: digits in factorial base representation of n become the exponents of successive prime factors whose product a(n) is.
+0
35
1, 2, 3, 6, 9, 18, 5, 10, 15, 30, 45, 90, 25, 50, 75, 150, 225, 450, 125, 250, 375, 750, 1125, 2250, 7, 14, 21, 42, 63, 126, 35, 70, 105, 210, 315, 630, 175, 350, 525, 1050, 1575, 3150, 875, 1750, 2625, 5250, 7875, 15750, 49, 98, 147, 294, 441, 882, 245, 490, 735, 1470, 2205, 4410, 1225, 2450, 3675, 7350, 11025, 22050, 6125, 12250, 18375, 36750, 55125, 110250, 343
COMMENTS
These are prime-factorization representations of single-variable polynomials where the coefficient of term x^(k-1) (encoded as the exponent of prime(k) in the factorization of n) is equal to the digit in one-based position k of the factorial base representation of n. See the examples.
FORMULA
Other identities.
For all n >= 0:
For all n >= 1:
(End)
EXAMPLE
n A007623 polynomial encoded as a(n) -------------------------------------------------------
0 0 0-polynomial (empty product) = 1
1 1 1*x^0 prime(1)^1 = 2
2 10 1*x^1 prime(2)^1 = 3
3 11 1*x^1 + 1*x^0 prime(2) * prime(1) = 6
4 20 2*x^1 prime(2)^2 = 9
5 21 2*x^1 + 1*x^0 prime(2)^2 * prime(1) = 18
6 100 1*x^2 prime(3)^1 = 5
7 101 1*x^2 + 1*x^0 prime(3) * prime(1) = 10
and:
23 321 3*x^2 + 2*x + 1 prime(3)^3 * prime(2)^2 * prime(1)
= 5^3 * 3^2 * 2 = 2250.
MATHEMATICA
a[n_] := Module[{k = n, m = 2, r, p = 2, q = 1}, While[{k, r} = QuotientRemainder[k, m]; k != 0|| r != 0, q *= p^r; p = NextPrime[p]; m++]; q]; Array[a, 100, 0] (* Amiram Eldar, Feb 07 2024 *)
CROSSREFS
Cf. A276078 (same terms in ascending order). Cf. also A000142, A001221, A001222, A002110, A007489, A008836, A019565, A033312, A034968, A048675, A051903, A059590, A060130, A076954, A246359, A248663, A262725, A276073, A276074, A351576, A351577, A351950, A351951, A351952, A351954.
Lexicographically earliest infinite sequence such that a(i) = a(j) => A083345(i) = A083345(j) and A343223(i) = A343223(j), for all i, j >= 1.
+0
2
1, 2, 2, 2, 2, 3, 2, 4, 5, 6, 2, 7, 2, 8, 9, 10, 2, 6, 2, 11, 12, 13, 2, 14, 5, 15, 2, 16, 2, 17, 2, 18, 19, 20, 21, 22, 2, 23, 24, 25, 2, 26, 2, 27, 28, 29, 2, 30, 10, 31, 32, 19, 2, 33, 24, 34, 35, 17, 2, 34, 2, 36, 25, 37, 38, 39, 2, 38, 40, 41, 2, 13, 2, 42, 43, 44, 45, 46, 2, 43, 47, 48, 2, 49, 35, 50, 51, 52, 2, 53, 54, 55, 56, 57, 58, 59, 2, 43, 60, 61
COMMENTS
Restricted growth sequence transform of the ordered pair [ A083345(n), A343223(n)].
PROG
(PARI)
up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om, invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om, invec[i], i); outvec[i] = u; u++ )); outvec; };
A083345(n) = { my(f=factor(n)); numerator(vecsum(vector(#f~, i, f[i, 2]/f[i, 1]))); };
A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
v374480 = rgs_transform(vector(up_to, n, Aux374480(n)));
a(n) = n' = arithmetic derivative of n: a(0) = a(1) = 0, a(prime) = 1, a(m*n) = m*a(n) + n*a(m).
(Formerly M3196)
+0
1065
0, 0, 1, 1, 4, 1, 5, 1, 12, 6, 7, 1, 16, 1, 9, 8, 32, 1, 21, 1, 24, 10, 13, 1, 44, 10, 15, 27, 32, 1, 31, 1, 80, 14, 19, 12, 60, 1, 21, 16, 68, 1, 41, 1, 48, 39, 25, 1, 112, 14, 45, 20, 56, 1, 81, 16, 92, 22, 31, 1, 92, 1, 33, 51, 192, 18, 61, 1, 72, 26, 59, 1, 156, 1, 39, 55, 80, 18, 71
COMMENTS
Can be extended to negative numbers by defining a(-n) = -a(n).
Based on the product rule for differentiation of functions: for functions f(x) and g(x), (fg)' = f'g + fg'. So with numbers, (ab)' = a'b + ab'. This implies 1' = 0. - Kerry Mitchell, Mar 18 2004 The derivative of a number x with respect to a prime number p as being the number "dx/dp" = (x-x^p)/p, which is an integer due to Fermat's little theorem. - Alexandru Buium, Mar 18 2004
The relation (ab)' = a'b + ab' implies 1' = 0, but it does not imply p' = 1 for p a prime. In fact, any function f defined on the primes can be extended uniquely to a function on the integers satisfying this relation: f(Product_i p_i^e_i) = (Product_i p_i^e_i) * (Sum_i e_i*f(p_i)/p_i). - Franklin T. Adams-Watters, Nov 07 2006 Let n be the product of a multiset P of k primes. Consider the k-dimensional box whose edges are the elements of P. Then the (k-1)-dimensional surface of this box is 2*a(n). For example, 2*a(25) = 20, the perimeter of a 5 X 5 square. Similarly, 2*a(18) = 42, the surface area of a 2 X 3 X 3 box. - David W. Wilson, Mar 11 2011 The arithmetic derivative n' was introduced, probably for the first time, by the Spanish mathematician José Mingot Shelly in June 1911 with "Una cuestión de la teoría de los números", work presented at the "Tercer Congreso Nacional para el Progreso de las Ciencias, Granada", cf. link to the abstract on Zentralblatt MATH, and L. E. Dickson, History of the Theory of Numbers. - Giorgio Balzarotti, Oct 19 2013 Sequence A157037 lists numbers with prime arithmetic derivative, i.e., indices of primes in this sequence. - M. F. Hasler, Apr 07 2015 Maybe the simplest "natural extension" of the arithmetic derivative, in the spirit of the above remark by Franklin T. Adams-Watters (2006), is the "pi based" version where f(p) = primepi(p), see sequence A258851. When f is chosen to be the identity map (on primes), one gets A066959. - M. F. Hasler, Jul 13 2015 When n is composite, it appears that a(n) has lower bound 2*sqrt(n), with equality when n is the square of a prime, and a(n) has upper bound (n/2)*log_2(n), with equality when n is a power of 2. - Daniel Forgues, Jun 22 2016 If n = p1*p2*p3*... where p1, p2, p3, ... are all the prime factors of n (not necessarily distinct), and h is a real number (we assume h nonnegative and < 1), the arithmetic derivative of n is equivalent to n' = lim_{h->0} ((p1+h)*(p2+h)*(p3+h)*... - (p1*p2*p3*...))/h. It also follows that the arithmetic derivative of a prime is 1. We could assume h = 1/N, where N is an integer; then the limit becomes {N -> oo}. Note that n = 1 is not a prime and plays the role of constant. - Giorgio Balzarotti, May 01 2023
REFERENCES
G. Balzarotti, P. P. Lava, La derivata aritmetica, Editore U. Hoepli, Milano, 2013.
E. J. Barbeau, Problem, Canad. Math. Congress Notes, 5 (No. 8, April 1973), 6-7.
L. E. Dickson, History of the Theory of Numbers, Vol. 1, Chapter XIX, p. 451, Dover Edition, 2005. (Work originally published in 1919.)
A. M. Gleason et al., The William Lowell Putnam Mathematical Competition: Problems and Solutions 1938-1964, Math. Assoc. America, 1980, p. 295.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
FORMULA
If n = Product p_i^e_i, a(n) = n * Sum (e_i/p_i).
a(n) = n * Sum_{p|n} v_p(n)/p, where v_p(n) is the largest power of the prime p dividing n. - Wesley Ivan Hurt, Jul 12 2015 For n >= 2, Sum_{k=2..n} floor(1/a(k)) = pi(n) = A000720(n) (see K. T. Atanassov article). - Ivan N. Ianakiev, Mar 22 2019 Limit_{n -> oo} (1/n^2)*Sum_{i=1..n} a(i) = A136141/2. Limit_{n -> oo} (1/n)*Sum_{i=1..n} a(i)/i = A136141. a(n) = n if and only if n = p^p, where p is a prime number. (End)
If n is not a prime, then a(n) >= 2*sqrt(n), or in other words, for all k >= 1 for which A002620(n)+k is not a prime, we have a( A002620(n)+k) > n. [See Ufnarovski and Åhlander, Theorem 9, point (3).] (End)
EXAMPLE
6' = (2*3)' = 2'*3 + 2*3' = 1*3 + 2*1 = 5.
Note that, for example, 2' + 3' = 1 + 1 = 2, (2+3)' = 5' = 1. So ' is not linear.
G.f. = x^2 + x^3 + 4*x^4 + x^5 + 5*x^6 + x^7 + 12*x^8 + 6*x^9 + 7*x^10 + ...
MAPLE
A003415 := proc(n) local B, m, i, t1, t2, t3; B := 1000000000039; if n<=1 then RETURN(0); fi; if isprime(n) then RETURN(1); fi; t1 := ifactor(B*n); m := nops(t1); t2 := 0; for i from 1 to m do t3 := op(i, t1); if nops(t3) = 1 then t2 := t2+1/op(t3); else t2 := t2+op(2, t3)/op(op(1, t3)); fi od: t2 := t2-1/B; n*t2; end;
local a, f;
a := 0 ;
for f in ifactors(n)[2] do
a := a+ op(2, f)/op(1, f);
end do;
n*a ;
MATHEMATICA
a[ n_] := If[ Abs @ n < 2, 0, n Total[ #2 / #1 & @@@ FactorInteger[ Abs @ n]]]; (* Michael Somos, Apr 12 2011 *) dn[0] = 0; dn[1] = 0; dn[n_?Negative] := -dn[-n]; dn[n_] := Module[{f = Transpose[FactorInteger[n]]}, If[PrimeQ[n], 1, Total[n*f[[2]]/f[[1]]]]]; Table[dn[n], {n, 0, 100}] (* T. D. Noe, Sep 28 2012 *)
PROG
(PARI) A003415(n) = {local(fac); if(n<1, 0, fac=factor(n); sum(i=1, matsize(fac)[1], n*fac[i, 2]/fac[i, 1]))} /* Michael B. Porter, Nov 25 2009 */ (PARI) apply( A003415(n)=vecsum([n/f[1]*f[2]|f<-factor(n+!n)~]), [0..99]) \\ M. F. Hasler, Sep 25 2013, updated Nov 27 2019 (PARI) a(n) = my(f=factor(n), r=[1/(e+!e)|e<-f[, 1]], c=f[, 2]); n*r*c; \\ Ruud H.G. van Tol, Sep 03 2023 (Haskell)
a003415 0 = 0
a003415 n = ad n a000040_list where
ad 1 _ = 0
ad n ps'@(p:ps)
| n < p * p = 1
| r > 0 = ad n ps
| otherwise = n' + p * ad n' ps' where
(n', r) = divMod n p
(Magma) Ad:=func<h | h*(&+[Factorisation(h)[i][2]/Factorisation(h)[i][1]: i in [1..#Factorisation(h)]])>; [n le 1 select 0 else Ad(n): n in [0..80]]; // Bruno Berselli, Oct 22 2013 (Python)
from sympy import factorint
return sum([int(n*e/p) for p, e in factorint(n).items()]) if n > 1 else 0
(Sage)
F = [] if n == 0 else factor(n)
return n * sum(g / f for f, g in F)
(GAP)
A003415:= Concatenation([0, 0], List(List([2..10^3], Factors),
(APL, Dyalog dialect) A003415 ← { ⍺←(0 1 2) ⋄ ⍵≤1:⊃⍺ ⋄ 0=(3⊃⍺)|⍵:((⊃⍺+(2⊃⍺)×(⍵÷3⊃⍺)) ((2⊃⍺)×(3⊃⍺)) (3⊃⍺)) ∇ ⍵÷3⊃⍺ ⋄ ((⊃⍺) (2⊃⍺) (1+(3⊃⍺))) ∇ ⍵} ⍝ Antti Karttunen, Feb 18 2024
CROSSREFS
Cf. A086134 (least prime factor of n'). Cf. A086131 (greatest prime factor of n'). Cf. A027471 (derivative of 3^(n-1)). Cf. A068237 (numerator of derivative of 1/n). Cf. A068238 (denominator of derivative of 1/n). Cf. A068328 (derivative of squarefree numbers). Cf. A260619 (derivative of hyperfactorial(n)). Cf. A260620 (derivative of superfactorial(n)). Cf. A068312 (derivative of triangular numbers). Cf. A068329 (derivative of Fibonacci(n)). Cf. A096371 (derivative of partition number). Cf. A099310 (derivative of phi(n)). Cf. A342925 (derivative of sigma(n)). Cf. A349905 (derivative of prime shift). Cf. A327860 (derivative of primorial base exp-function). Cf. A369252 (derivative of products of three odd primes), A369251 (same sorted). Cf. A068346 (second derivative of n). Cf. A099306 (third derivative of n). Cf. A258644 (fourth derivative of n). Cf. A258645 (fifth derivative of n). Cf. A258646 (sixth derivative of n). Cf. A258647 (seventh derivative of n). Cf. A258648 (eighth derivative of n). Cf. A258649 (ninth derivative of n). Cf. A258650 (tenth derivative of n). Cf. A185232 (n-th derivative of n). Cf. A258651 (A(n,k) = k-th arithmetic derivative of n). Cf. A098699 (least x such that x' = n, antiderivative of n). Cf. A098700 (n such that x' = n has no integer solution). Cf. A099302 (number of solutions to x' = n). Cf. A099303 (greatest x such that x' = n). Cf. A099304 (least k such that (n+k)' = n' + k'). Cf. A099305 (number of solutions to (n+k)' = n' + k'). Cf. A328235 (least k > 0 such that (n+k)' = u * n' for some natural number u). Cf. A328236 (least m > 1 such that (m*n)' = u * n' for some natural number u). Cf. A099307 (least k such that the k-th arithmetic derivative of n is zero). Cf. A099308 (k-th arithmetic derivative of n is zero for some k). Cf. A099309 (k-th arithmetic derivative of n is nonzero for all k). Cf. A129150 (n-th derivative of 2^3). Cf. A129151 (n-th derivative of 3^4). Cf. A129152 (n-th derivative of 5^6). Cf. A189481 (x' = n has a unique solution). Cf. A229501 (n divides the n-th partial sum). Cf. A328303 (n' is not squarefree), A328252 (n' is squarefree, but n is not). Cf. A328248 (least k such that the (k-1)-th derivative of n is squarefree). Cf. A328251 (k-th arithmetic derivative is never squarefree for any k >= 0). Cf. A256750 (least k such that the k-th derivative is either 0 or has a factor p^p). Cf. A327928 (number of distinct primes p such that p^p divides n'). Cf. A342003 (max. exponent k for any prime power p^k that divides n'). Cf. A327929 (n' has at least one divisor of the form p^p). Cf. A327978 (n' is primorial number > 1). Cf. A328243 (n' is a partial sum of primorial numbers and larger than one). Cf. A328310 (maximal prime exponent of n' minus maximal prime exponent of n). Cf. A328320 (max. prime exponent of n' is less than that of n). Cf. A328321 (max. prime exponent of n' is >= that of n). Cf. A328383 (least k such that the k-th derivative of n is either a multiple or a divisor of n, but not both). Cf. A263111 (the ordinal transform of a). Cf. A305809 (Dirichlet convolution square). Cf. A069359 (similar formula which agrees on squarefree numbers). Cf. A258851 (the pi-based arithmetic derivative of n). Cf. A038554 (another sequence using "derivative" in its name, but involving binary expansion of n).
Numbers k for which k' / gcd(k,k') is even, where k' stands for the arithmetic derivative of k, A003415.
+0
20
1, 9, 12, 15, 16, 20, 21, 25, 28, 33, 35, 39, 44, 49, 51, 52, 55, 57, 65, 68, 69, 76, 77, 81, 85, 87, 91, 92, 93, 95, 108, 111, 115, 116, 119, 121, 123, 124, 129, 133, 135, 141, 143, 144, 145, 148, 155, 159, 161, 164, 169, 172, 177, 180, 183, 185, 187, 188, 189, 192, 201, 203, 205, 209, 212, 213, 215, 217, 219, 221
COMMENTS
Numbers k for which A276085(k) is a multiple of four. Even terms in this sequence are all multiples of four.
A multiplicative semigroup; if m and n are in the sequence then so is m*n.
(End)
Appears to be products of an even number of terms from {4} U A065091 (counting repetitions). - Peter Munn, Jul 15 2024
CROSSREFS
Positions of even terms in A083345.
Non-multiples of 3 whose arithmetic derivative, or equally, the sum of prime factors (with multiplicity) is a multiple of 3.
+0
8
1, 8, 14, 20, 26, 35, 38, 44, 50, 62, 64, 65, 68, 74, 77, 86, 92, 95, 110, 112, 116, 119, 122, 125, 134, 143, 146, 155, 158, 160, 161, 164, 170, 185, 188, 194, 196, 203, 206, 208, 209, 212, 215, 218, 221, 230, 236, 242, 254, 275, 278, 280, 284, 287, 290, 299, 302, 304, 305, 314, 323, 326, 329, 332, 335, 341, 343
COMMENTS
This is a subsequence of A373475, containing all its terms that are not multiples of 3. (See comments in A373475 for a proof). The first difference from A373475 is at n=4186, where A373475(4186) = 19683 = 3^9, the value which is missing from this sequence. - Antti Karttunen, Jun 07 2024 A multiplicative semigroup: if m and n are in the sequence, then so is m*n.
Numbers that are not multiples of 3, and the multiplicities of prime factors of the forms 3m+1 ( A002476) and 3m-1 ( A003627) are equal modulo 3. Like A373597, which is a subsequence, also this sequence can be viewed as a kind of k=3 variant of A046337. A289142, numbers whose sum of prime factors (with multiplicity, A001414) is a multiple of 3, is generated (as a multiplicative semigroup) by the union of this sequence with {3}.
A327863, numbers whose arithmetic derivative is a multiple of 3, is generated by this sequence and A008591.
A373478, numbers that are in the intersection of A289142 and A327863, is generated by the union of this sequence with {9, 27}.
A373475, numbers that are in the intersection of A289142 and A369644 (positions of multiples of 3 in A083345), is generated by the union of this sequence with {19683}, where 19683 = 3^9.
(End)
The integers in the multiplicative subgroup of positive rationals generated by semiprimes of the form 3m+2 ( A344872) and cubes of primes except 27. - Peter Munn, Jun 19 2024
EXAMPLE
280 = 2*2*2*5*7 is included as it is not a multiple of 3, and one of its prime factors (7) is of the form 3m+1 and four are of the form 3m-1, and because 4 == 1 (mod 3). Also, A001414(280) = 18, and A003415(280) = 516, both of which are multiples of 3. - Antti Karttunen, Jun 12 2024
CROSSREFS
Cf. also A046337, A360110, A369969 for cases k=2, 4, 5 of "Nonmultiples of k whose arithmetic derivative is a multiple of k".
EXTENSIONS
Name amended with an alternative definition by Antti Karttunen, Jun 11 2024
Lexicographically earliest infinite sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = [ A003415(n), A085731(n), A007814(n), A007949(n)], for all i, j >= 1.
+0
2
1, 2, 3, 4, 5, 6, 5, 7, 8, 9, 5, 10, 5, 11, 12, 13, 5, 14, 5, 15, 16, 17, 5, 18, 19, 20, 21, 22, 5, 23, 5, 24, 25, 26, 27, 28, 5, 29, 30, 31, 5, 32, 5, 33, 34, 35, 5, 36, 37, 38, 39, 40, 5, 41, 42, 43, 44, 45, 5, 46, 5, 47, 48, 49, 50, 51, 5, 52, 53, 54, 5, 55, 5, 56, 57, 58, 50, 59, 5, 60, 61, 62, 5, 63, 64, 65, 66, 67, 5, 68, 69, 70, 71, 72, 73, 74, 5, 75
COMMENTS
For all i, j >= 1:
a(i) = a(j) => b(i) = b(j), where b can be any of the sequences listed at the crossrefs-section, under "some of the other matched sequences".
PROG
(PARI)
up_to = 100000;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om, invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om, invec[i], i); outvec[i] = u; u++ )); outvec; };
A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
Aux374040(n) = { my(d= A003415(n)); [d, gcd(n, d), valuation(n, 2), valuation(n, 3)]; }; v374040 = rgs_transform(vector(up_to, n, Aux374040(n)));
CROSSREFS
Differs from A305900 first at n=77, where a(77) = 50, while A305900(77) = 59.
Lexicographically earliest infinite sequence such that for all i, j >= 1, a(i) = a(j) => f(i) = f(j), where f(1) = 1, and for n > 1, f(n) = [ A083345(n), A374132(n), A374133(n)], where A083345 is the numerator of the fully additive function with a(p) = 1/p, and A374132 and A374133 are the 2- and 3-adic valuations of A276085, which is fully additive with a(p) = p#/p.
+0
4
1, 2, 3, 3, 4, 5, 4, 6, 7, 8, 4, 9, 4, 10, 11, 7, 4, 8, 4, 12, 13, 14, 4, 15, 16, 17, 4, 18, 4, 19, 4, 20, 21, 22, 23, 24, 4, 25, 26, 27, 4, 28, 4, 29, 30, 31, 4, 32, 16, 10, 33, 34, 4, 35, 36, 37, 38, 39, 4, 40, 4, 41, 42, 43, 44, 45, 4, 46, 47, 48, 4, 14, 4, 49, 50, 33, 51, 52, 4, 50, 53, 54, 4, 55, 56, 57, 58, 59, 4, 60, 61, 62, 63, 64, 65, 66, 4, 15, 67, 68
COMMENTS
Restricted growth sequence transform of the function f given in the definition.
For all i, j >= 1:
PROG
(PARI)
up_to = 100000;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om, invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om, invec[i], i); outvec[i] = u; u++ )); outvec; };
A083345(n) = { my(f=factor(n)); numerator(vecsum(vector(#f~, i, f[i, 2]/f[i, 1]))); };
A276085(n) = { my(f=factor(n)); sum(k=1, #f~, f[k, 2]*prod(i=1, primepi(f[k, 1]-1), prime(i))); };
Aux374131(n) = if(1==n, n, my(u= A276085(n)); [ A083345(n), valuation(u, 2), valuation(u, 3)]); v374131 = rgs_transform(vector(up_to, n, Aux374131(n)));
Primorial base log-function: fully additive with a(p) = p#/p, where p# = A034386(p).
+0
146
0, 1, 2, 2, 6, 3, 30, 3, 4, 7, 210, 4, 2310, 31, 8, 4, 30030, 5, 510510, 8, 32, 211, 9699690, 5, 12, 2311, 6, 32, 223092870, 9, 6469693230, 5, 212, 30031, 36, 6, 200560490130, 510511, 2312, 9, 7420738134810, 33, 304250263527210, 212, 10, 9699691, 13082761331670030, 6, 60, 13, 30032, 2312, 614889782588491410, 7, 216, 33, 510512, 223092871, 32589158477190044730, 10
COMMENTS
This is a left inverse of A276086 ("primorial base exp-function"), hence the name "primorial base log-function". When the domain is restricted to the terms of A048103, this works also as a right inverse, as A276086(a( A048103(n))) = A048103(n) for all n >= 1. - Antti Karttunen, Apr 24 2022
FORMULA
a(1) = 0, a(n) = (e1* A002110(i1-1) + ... + ez* A002110(iz-1)) when n = prime(i1)^e1 * ... * prime(iz)^ez. Other identities.
For all n >= 0:
a( A108951(n)) = A346105(n). [The latter has a similar additive formula as this sequence, but instead of primorials, uses their partial sums] When applied to sequences where a certain subset of the divisors of n has been multiplicatively encoded with the help of A276086, this yields a corresponding number-theoretical sequence, i.e. completes their computation: In the following group, the sum of the rhs-sequences is n [on each row, as say, A328841(n)+ A328842(n)=n], because the pointwise product of the corresponding lhs-sequences is A276086: The sum or difference of the rhs-sequences is A108951: Here the two sequences are inverse permutations of each other:
Other correspondences:
a( A276076(n)) = A351576(n). [Sequence reinterpreting factorial base representation as a primorial base representation] (End)
MATHEMATICA
nn = 60; b = MixedRadix[Reverse@ Prime@ Range@ PrimePi[nn + 1]]; Table[FromDigits[#, b] &@ Reverse@ If[n == 1, {0}, Function[k, ReplacePart[Table[0, {PrimePi[k[[-1, 1]]]}], #] &@ Map[PrimePi@ First@ # -> Last@ # &, k]]@ FactorInteger@ n], {n, nn}] (* Version 10.2, or *)
f[w_List] := Total[Times @@@ Transpose@ {Map[Times @@ # &, Prime@ Range@ Range[0, Length@ w - 1]], Reverse@ w}]; Table[f@ Reverse@ If[n == 1, {0}, Function[k, ReplacePart[Table[0, {PrimePi[k[[-1, 1]]]}], #] &@ Map[PrimePi@ First@ # -> Last@ # &, k]]@ FactorInteger@ n], {n, 60}] (* Michael De Vlieger, Aug 30 2016 *)
PROG
(Scheme, with memoization-macro definec)
(PARI) A276085(n) = { my(f=factor(n)); sum(k=1, #f~, f[k, 2]*prod(i=1, primepi(f[k, 1]-1), prime(i))); }; \\ Antti Karttunen, Mar 15 2021, Jun 23 2024 (Python)
from sympy import primorial, primepi, factorint
def a002110(n):
return 1 if n<1 else primorial(n)
def a(n):
f=factorint(n)
return sum(f[i]*a002110(primepi(i) - 1) for i in f)
CROSSREFS
Cf. A000040, A000720, A002110, A028234, A034386, A048103, A049345, A055396, A067029, A108951, A143293, A276154, A328316, A328624, A328625, A328768, A328832, A346105, A351576. Cf. A373145 [= gcd( A003415(n), a(n))], A373361 [= gcd(n, a(n))], A373362 [= gcd( A001414(n), a(n))], A373485 [= gcd( A083345(n), a(n))], A373835 [= gcd(bigomega(n), a(n))], and also A373367 and A373147 [= A003415(n) mod a(n)], A373148 [= a(n) mod A003415(n)]. Other completely additive sequences with primes p mapped to a function of p include: A001222 (with a(p)=1), A001414 (with a(p)=p), A059975 (with a(p)=p-1), A341885 (with a(p)=p*(p+1)/2), A373149 (with a(p)=prevprime(p)), A373158 (with a(p)=p#).
EXTENSIONS
Name simplified, the old name moved to the comments - Antti Karttunen, Jun 23 2024
Numbers k for which A059975(k) and A083345(k) are both multiples of 3, where A059975 is fully additive with a(p) = p-1, and A083345 is the numerator of the fully additive function with a(p) = 1/p.
+0
5
1, 8, 20, 44, 50, 64, 68, 92, 110, 116, 125, 160, 164, 170, 188, 212, 230, 236, 242, 275, 284, 290, 332, 343, 352, 356, 374, 400, 404, 410, 425, 428, 452, 470, 506, 512, 524, 530, 544, 548, 575, 578, 590, 596, 605, 637, 638, 668, 692, 710, 716, 725, 736, 764, 782, 788, 830, 880, 890, 902, 908, 928, 931, 932, 935
COMMENTS
A multiplicative semigroup: if m and n are in the sequence, then so is m*n.
CROSSREFS
Positions of multiples of 3 in A373377.
a(n) = 1 if A059975(n) and A083345(n) are both multiples of 3, otherwise 0, where A059975 is fully additive with a(p) = p-1, and A083345 is the numerator of the fully additive function with a(p) = 1/p.
+0
6
1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1
FORMULA
a(n) = [ A373377(n) == 0 (mod 3)], where [ ] is the Iverson bracket.
PROG
(PARI)
A059975(n) = {my(f = factor(n)); sum(i = 1, #f~, f[i, 2]*(f[i, 1] - 1)); };
A083345(n) = { my(f=factor(n)); numerator(vecsum(vector(#f~, i, f[i, 2]/f[i, 1]))); };
CROSSREFS
Characteristic function of A373492.
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