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 A174961 #27 Apr 09 2025 07:24:18
%S A174961 1,2,1,2,3,2,2,4,1,2,2,4,5,4,2,2,6,1,2,2,4,4,4,2,5,2,8,2,2,6,3,4,4,4,
%T A174961 2,8,2,2,5,4,8,6,2,2,8,1,2,2,4,6,4,4,4,4,11,2,2,4,4,2,4,8,6,2,8,1,2,2,
%U A174961 2,6,10,4,2,4,10,5,4,8,4,2,6,2,12,4,8,5,4,4,4,2,12,2,4,2
%N A174961 Number of non-unitary divisors of the n-th nonsquarefree number.
%C A174961 The nonzero terms of A048105.
%C A174961 Also number of nonsquarefree divisors of the n-th nonsquarefree number. The terms in A013929 which correspond to records in this sequence are given by A309141(n); n >= 2. - _David James Sycamore_, Jan 07 2025
%H A174961 Amiram Eldar, <a href="/A174961/b174961.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from G. C. Greubel)
%F A174961 From _Amiram Eldar_, Dec 09 2023: (Start)
%F A174961 a(n) = A048105(A013929(n)).
%F A174961 Sum_{k=1..n} a(k) ~ (n/zeta(2)) * (log(n) + 2*gamma - 1 - 2*zeta'(2)/zeta(2)), where gamma is Euler's constant (A001620). (End)
%F A174961 a(n) = A000005(A013929(n)) - A000005(A007947(A013929(n))). - _David James Sycamore_, Jan 07 2025
%e A174961 For n = 4, the fourth nonsquarefree number is A013929(4) = 12 which has 2 non-unitary divisors, 2 and 6. Therefore a(4) = 2.
%e A174961 The number of nonsquarefree divisors of 12 is also = 2 (4 and 12). For n = 55, A013929(55) = 144 so by the third formula above a(55) = A000005(144) - A000005(6) = 15 - 4 = 11 = number of nonsquarefree divisors of 144 (4,8,9,12,16,18,24,36,48,72,144). - _David James Sycamore_, Jan 07 2025
%t A174961 Select[Table[DivisorSigma[0, n] - 2^(PrimeNu[n]), {n, 1, 500}], # > 0 &] (* _G. C. Greubel_, May 21 2017 *)
%o A174961 (PARI) lista(kmax) = {my(f); for(k = 1, kmax, f = factor(k); if(!issquarefree(f), print1(numdiv(f) - 2^omega(f), ", ")));} \\ _Amiram Eldar_, Dec 09 2023
%o A174961 (Python)
%o A174961 from math import prod, isqrt
%o A174961 from sympy import mobius, factorint
%o A174961 def A174961(n):
%o A174961 def f(x): return n+sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))
%o A174961 m, k = n, f(n)
%o A174961 while m != k:
%o A174961 m, k = k, f(k)
%o A174961 return -(1<<len(g:=factorint(m).values()))+prod(e+1 for e in g) # _Chai Wah Wu_, Aug 12 2024
%Y A174961 Cf. A013929, A048105.
%Y A174961 Cf. A001620, A013661, A306016.
%Y A174961 Cf. A034444, A309141.
%K A174961 nonn
%O A174961 1,2
%A A174961 N. Wu (neil_wu0626(AT)yahoo.com), Apr 02 2010
%E A174961 Edited by _Amiram Eldar_, Dec 09 2023