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

1,2

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

Links

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

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

Example

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

Mathematica

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

Prog

(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<<len(g:=factorint(m).values()))+prod(e+1 for e in g) # Chai Wah Wu, Aug 12 2024

Crossrefs

Cf. A013929, A048105.

Cf. A001620, A013661, A306016.

Cf. A034444, A309141.

Sequence in context: A283845 A365543 A058071 * A104889 A356122 A290979

Adjacent sequences: A174958 A174959 A174960 * A174962 A174963 A174964

Keyword

nonn,changed

Author

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

Extensions

Edited by Amiram Eldar, Dec 09 2023

Status

approved