A126706 - OEIS

A126706

Positive integers which are neither squarefree integers nor prime powers.

287

12, 18, 20, 24, 28, 36, 40, 44, 45, 48, 50, 52, 54, 56, 60, 63, 68, 72, 75, 76, 80, 84, 88, 90, 92, 96, 98, 99, 100, 104, 108, 112, 116, 117, 120, 124, 126, 132, 135, 136, 140, 144, 147, 148, 150, 152, 153, 156, 160, 162, 164, 168, 171, 172, 175, 176, 180, 184, 188

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OFFSET

1,1

LINKS

Michael De Vlieger, Table of n, a(n) for n = 1..10000

EXAMPLE

45 is in the sequence because 45=3^2*5, i.e., neither squarefree nor a prime power.

MAPLE

with(numtheory): a:=proc(n) if mobius(n)=0 and nops(factorset(n))>1 then n else fi end: seq(a(n), n=1..230); # Emeric Deutsch, Feb 17 2007

MATHEMATICA

Select[Range[200], Max @@ Last /@ FactorInteger[ # ] >1 && Length[FactorInteger[ # ]] > 1 &] (* Ray Chandler, Feb 17 2007 *)

Select[Range[200], !SquareFreeQ[#]&&!PrimePowerQ[#]&] (* Harvey P. Dale, Aug 05 2023 *)

PROG

(PARI) isok(k) = !issquarefree(k) && !isprimepower(k); \\ Michel Marcus, Nov 02 2022

(Python)

from math import isqrt

from sympy import primepi, integer_nthroot, mobius

def A126706(n):

def f(x): return int(n+sum(primepi(integer_nthroot(x, k)[0]) for k in range(2, x.bit_length()))+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 m # Chai Wah Wu, Aug 15 2024

CROSSREFS

Cf. A005117, A000961, A059404, A355447 (characteristic function).

Disjoint union of A360765 and A363082.

Positions of nonzero terms in A389171.

Subsequence of A389462.

Sequence in context: A387723 A389812 A342973 * A123711 A200511 A370409

Adjacent sequences: A126703 A126704 A126705 * A126707 A126708 A126709

KEYWORD

nonn

AUTHOR

Leroy Quet, Feb 11 2007

EXTENSIONS

Extended by Emeric Deutsch and Ray Chandler, Feb 17 2007

STATUS

approved