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 A361430 #27 Apr 15 2025 11:56:41
%S A361430 1,0,0,1,0,0,0,2,1,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,1,0,2,0,0,0,0,4,0,0,
%T A361430 0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,
%U A361430 0,0,0,2,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0
%N A361430 Multiplicative with a(p^e) = e - 1.
%C A361430 a(n) is the number of coreful divisors d of n such that n/d is also a coreful divisor of n (a coreful divisor d of a number n is a divisor with the same set of distinct prime factors as n, see A307958). - _Amiram Eldar_, Aug 15 2023
%H A361430 Amiram Eldar, <a href="/A361430/b361430.txt">Table of n, a(n) for n = 1..10000</a>
%H A361430 Vaclav Kotesovec, <a href="/A361430/a361430.jpg">Graph - the asymptotic ratio (10^9 terms)</a>
%F A361430 Dirichlet g.f.: Product_{primes p} (1 + 1/(p^s - 1)^2).
%F A361430 Dirichlet g.f.: zeta(2*s) * zeta(3*s)^2 * Product_{primes p} (1 + 2/p^(4*s) + 2/p^(5*s) - 1/p^(6*s) - 2/p^(7*s) - 2/p^(8*s)).
%F A361430 Let f(s) = Product_{primes p} (1 + 2/p^(4*s) + 2/p^(5*s) - 1/p^(6*s) - 2/p^(7*s) - 2/p^(8*s)), then
%F A361430 Sum_{k=1..n} a(k) ~ f(1/2) * zeta(3/2)^2 * sqrt(n) + zeta(2/3) * (f(1/3) * (log(n) + 6*gamma - 3 + 2*zeta'(2/3)/zeta(2/3)) + f'(1/3)) * n^(1/3) / 3, where
%F A361430 f(1/2) = Product_{primes p} (1 + 2/p^2 + 2/p^(5/2) - 1/p^3 - 2/p^(7/2) - 2/p^4) = 2.20286226691565931157047065666916419062717171359087693723221239...
%F A361430 f(1/3) = Product_{primes p} (1 + 2/p^(4/3) + 2/p^(5/3) - 1/p^2 - 2/p^(7/3) - 2/p^(8/3)) = 6.250573144372477079986352917664218040797528021629950408099536...
%F A361430 f'(1/3) = f(1/3) * Sum_{primes p} (-2*(-8 + p^(1/3) + 4*p^(2/3)) * log(p) / (-2 + p^(2/3) + p - p^(5/3) + p^2)) = -90.898558294301467740374653006294640945295... and gamma is the Euler-Mascheroni constant A001620.
%F A361430 Conjecture: a(n) = abs(A298826(n)).
%F A361430 a(n) > 0 if and only if n is powerful (A001694). - _Amiram Eldar_, Aug 15 2023
%t A361430 g[p_, e_] := e-1; a[1] = 1; a[n_] := Times @@ g @@@ FactorInteger[n]; Array[a, 200]
%o A361430 (PARI) for(n=1, 100, print1(direuler(p=2, n, 1 + 1/(1 - 1/X)^2)[n], ", "))
%o A361430 (PARI) a(n) = my(f=factor(n)); for (k=1, #f~, f[k,1] = f[k,2]-1; f[k,2] = 1); factorback(f); \\ _Michel Marcus_, Mar 13 2023
%o A361430 (Python)
%o A361430 from math import prod
%o A361430 from sympy import factorint
%o A361430 def A361430(n): return prod(e-1 for e in factorint(n).values()) # _Chai Wah Wu_, Apr 15 2025
%Y A361430 Cf. A001694, A298826, A307958, A335850 (indices of records).
%Y A361430 Cf. A005361 (multiplicative with a(p^e) = e), A000005 (e+1), A343443 (e+2), A360997 (e+3), A360908 (2*e-1), A360910 (3*e-1), A360911 (3*e-2).
%K A361430 nonn,easy,mult
%O A361430 1,8
%A A361430 _Vaclav Kotesovec_, Mar 11 2023