A174961 Number of non-unitary divisors of the n-th nonsquarefree number.

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, 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, 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

Offset: 1

Author

N. Wu (neil_wu0626(AT)yahoo.com), Apr 02 2010

Keywords

• nonn

Comments

The nonzero terms of A048105.

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

Examples

			For n = 4, the fourth nonsquarefree number is A013929(4) = 12 which has 2 non-unitary divisors, 2 and 6. Therefore a(4) = 2.
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
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from G. C. Greubel)

Crossrefs

Cf. A013929, A048105.

Cf. A001620, A013661, A306016.

Cf. A034444, A309141.

Programs

Mathematica

Select[Table[DivisorSigma[0, n] - 2^(PrimeNu[n]), {n, 1, 500}], # > 0 &] (* G. C. Greubel, May 21 2017 *)

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

Python

from math import prod, isqrt
from sympy import mobius, factorint
def A174961(n):
    def f(x): return n+sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    return -(1<Chai Wah Wu, Aug 12 2024

Formula

From Amiram Eldar, Dec 09 2023: (Start)

a(n) = A048105(A013929(n)).

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)

a(n) = A000005(A013929(n)) - A000005(A007947(A013929(n))). - David James Sycamore, Jan 07 2025

Extensions

Edited by Amiram Eldar, Dec 09 2023