A212177 - OEIS

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,13

COMMENTS

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

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).

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

EXAMPLE

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

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 *)

PROG

(Haskell)

a212177 n = a212177_list !! (n-1)

a212177_list = filter (> 0) a056170_list

-- Reinhard Zumkeller, Dec 29 2012

(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: