A212177 Number of exponents >= 2 in the canonical prime factorization of the n-th nonsquarefree number (A013929(n)).

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1

Offset: 1

Author

Matthew Vandermast, Jun 03 2012

Keywords

• nonn
• easy

Comments

Length of second signature of A013929(n) (cf. A212172).

Examples

			24 = 2^3*3 has 1 exponent of size 2 or greater in its prime factorization. Since 24 = A013929(8), a(8) = 1.
		

References

• M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 844.

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
• M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
• Primefan, The First 2500 Integers Factored (1st of 5 pages).

Crossrefs

Cf. A013929, A212172, A212174.

Cf. A046028, A056170, A085548, A124010, A229099.

Programs

Haskell

a212177 n = a212177_list !! (n-1)
a212177_list = filter (> 0) a056170_list
-- Reinhard Zumkeller, Dec 29 2012

Mathematica

f[n_] := Module[{c = Count[FactorInteger[n][[;; , 2]], ?(# > 1&)]}, If[n > 1 && c > 0, c, Nothing]]; f[1] = 0; Array[f, 300] (* _Amiram Eldar, Oct 01 2023 *)

Python

from math import isqrt
from sympy import mobius, factorint
def A212177(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 sum(1 for e in factorint(m).values() if e>1) # Chai Wah Wu, Jul 22 2024

Formula

a(n) = A056170(A013929(n)).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = (Sum_{p prime} 1/p^2)/(1-1/zeta(2)) = A085548 / A229099 = 1.15347789194214704903... . - Amiram Eldar, Oct 01 2023