This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A112624 #70 Mar 08 2024 08:07:06
%S A112624 1,1,1,2,1,1,1,6,2,1,1,2,1,1,1,24,1,2,1,2,1,1,1,6,2,1,6,2,1,1,1,120,1,
%T A112624 1,1,4,1,1,1,6,1,1,1,2,2,1,1,24,2,2,1,2,1,6,1,6,1,1,1,2,1,1,2,720,1,1,
%U A112624 1,2,1,1,1,12,1,1,2,2,1,1,1,24,24,1,1,2,1,1,1,6,1,2,1,2,1,1,1,120,1,2,2,4,1
%N A112624 If p^b(p,n) is the highest power of the prime p dividing n, then a(n) = Product_{p|n} b(p,n)!.
%C A112624 The logarithm of the Dirichlet series with the reciprocals of this sequence as coefficients is the Dirichlet series with the characteristic function of primes A010051 as coefficients. - _Mats Granvik_, Apr 13 2011
%H A112624 Antti Karttunen, <a href="/A112624/b112624.txt">Table of n, a(n) for n = 1..10000</a>
%H A112624 Rafael Jakimczuk, <a href="http://dx.doi.org/10.13140/RG.2.2.27745.48487">Generalizations of Mertens's Formula and k-Free and s-Full Numbers with Prime Divisors in Arithmetic Progression</a>, ResearchGate, 2024.
%H A112624 <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>.
%F A112624 From _Antti Karttunen_, May 29 2017: (Start)
%F A112624 a(1) = 1 and for n > 1, a(n) = A000142(A067029(n)) * a(A028234(n)).
%F A112624 a(n) = A246660(A156552(n)). (End)
%F A112624 From _Mats Granvik_, Mar 05 2019: (Start)
%F A112624 log(a(n)) = inverse Möbius transform of log(A306694(n)).
%F A112624 log(a(n)) = Sum_{k=1..n} [k|n]*log(A306694(n/k))*A000012(k). (End)
%F A112624 From _Amiram Eldar_, Mar 08 2024: (Start)
%F A112624 Let f(n) = 1/a(n). Formulas from Jakimczuk (2024, pp. 12-15):
%F A112624 Dirichlet g.f. of f(n): Sum_{n>=1} f(n)/n^s = exp(P(s)), where P(s) is the prime zeta function.
%F A112624 Sum_{k=1..n} f(k) = c * n + o(n), where c = A240953.
%F A112624 Sum_{k=1..n} f(k)/k = c * log(n) + o(log(n)), where c = A240953. (End)
%e A112624 45 = 3^2 * 5^1. So a(45) = 2! * 1! = 2.
%p A112624 w := n -> op(2, ifactors(n)): a := n -> mul(factorial(w(n)[j][2]), j = 1..nops(w(n))): seq(a(n), n = 1..101); # Emeric Deutsch, May 17 2012
%t A112624 f[n_] := Block[{fi = Last@Transpose@FactorInteger@n}, Times @@ (fi!)]; Array[f, 101] (* _Robert G. Wilson v_, Dec 27 2005 *)
%o A112624 (PARI) A112624(n) = { my(f = factor(n), m = 1); for (k=1, #f~, m *= f[k, 2]!; ); m; } \\ _Antti Karttunen_, May 28 2017
%o A112624 (Sage)
%o A112624 def A112624(n):
%o A112624 return mul(factorial(s[1]) for s in factor(n))
%o A112624 [A112624(i) for i in (1..101)] # _Peter Luschny_, Jun 15 2013
%o A112624 (Scheme) (define (A112624 n) (if (= 1 n) n (* (A000142 (A067029 n)) (A112624 (A028234 n))))) ;; _Antti Karttunen_, May 29 2017
%Y A112624 For row > 1: a(n) = row products of A100995(A126988), when neglecting zero elements.
%Y A112624 Cf. A000012, A000142, A010051, A028234, A067029, A112622, A112623, A156552, A240953, A246660, A306694.
%K A112624 nonn,easy,mult
%O A112624 1,4
%A A112624 _Leroy Quet_, Dec 25 2005
%E A112624 More terms from _Robert G. Wilson v_, Dec 27 2005