A024619 Numbers that are not powers of primes p^k (k >= 0); complement of A000961.

6, 10, 12, 14, 15, 18, 20, 21, 22, 24, 26, 28, 30, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99, 100, 102, 104, 105, 106, 108, 110, 111, 112

Offset: 1

Author

Clark Kimberling

Keywords

• nonn
• easy

Comments

The sequence of numbers divisible by a prime number of primes coincides with this up to 210, which has 4 prime factors. - Lior Manor, Aug 23 2001

A085970(n) = Max{k: a(k)<=n}.

Numbers n such that LCM of proper divisors of n equals neither 1 nor n. - Labos Elemer, Dec 01 2004

a(n) provides bases b in which automorphic numbers m^2 ending with m in base b exist. In the complement there aren't any automorphic numbers. - Martin Renner, Dec 07 2011

Numbers with at least 2 distinct prime factors. - Jonathan Sondow, Oct 17 2013

There exists an equiangular n-gon whose edge lengths form a permutation of 1, 2, ..., n if and only if n is in the sequence (see Woeginger's survey and Munteanu & Munteanu). - Jonathan Sondow, Oct 17 2013

Numbers that are the product of two relatively prime factors. These numbers are used in testing a sequence for multiplicativity. - Michael Somos, Jun 02 2015

A theorem from Donald McCarthy: Let d be any positive integer which is not a prime power; then there exists a finite group whose order is divisible by d but which contains no subgroup of order d (see link and A340511). - Bernard Schott, Dec 04 2021

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000 (first 8719 terms from Daniel Forgues)
• Donald McCarthy, Sylow's theorem is a sharp partial converse to Lagrange's theorem, Mathematische Zeitschrift, 113, 383-384 (1970).
• Marius Munteanu and Laura Munteanu, Rational equiangular polygons, Applied Math., 4 (2013), 1460-1465.
• Laurentiu Panaitopol, Some of the properties of the sequence of powers of prime numbers, Rocky Mountain Journal of Mathematics, Volume 31, Number 4, Winter 2001.
• Eric Weisstein's World of Mathematics, Prime Power
• Wikipedia, Prime power
• G. J. Woeginger, Nothing new about equiangular polygons, Amer. Math. Monthly, 120 (2013), 849-850.
• Günter Ziegler and Brady Haran, Cannons and Sparrows, Numberphile video (2018).

Crossrefs

Cf. A000040, A000961 (complement), A001221, A014963, A020500, A085970.

Cf. A340511.

Subsequence of A080257.

Programs

Haskell

a024619 n = a024619_list !! (n-1)
a024619_list = filter ((== 0) . a010055) [1..]
-- Reinhard Zumkeller, Nov 17 2011

Magma

IsA024619:=func< n | not IsPrime(n) and not (t and IsPrime(b) where t, b, A024619(n)%20%5D;%20//%20_Klaus%20Brockhaus">:=IsPower(n)) >; [ n: n in [2..200] | IsA024619(n) ]; // _Klaus Brockhaus, Feb 25 2011

Maple

a := proc(n) numtheory[factorset](n); if 1 < nops(%) then n else NULL fi end:
seq(a(i), i=1..110); # Peter Luschny, Aug 11 2009

Mathematica

Select[Range@111, Length@FactorInteger@# > 1 &] (* Robert G. Wilson v, Dec 07 2005 *)

PARI

is(n)=n>5 && !isprimepower(n) \\ Charles R Greathouse IV, Mar 21 2013

Python

from sympy import primepi
from sympy.ntheory.primetest import integer_nthroot
def A024619(n):
    def f(x): return int(n+1+sum(primepi(integer_nthroot(x,k)[0]) for k in range(1,x.bit_length())))
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    return m # Chai Wah Wu, Jul 23 2024

Sage

def A024619_list(n) :
    return [k for k in (2..n) if not k.is_prime() and not k.is_prime_power()]
A024619_list(112)  # Peter Luschny, Feb 03 2012 [corrected by Terry D. Grant, Sep 16 2020]

Formula

A001221(a(n)) > 1.

A014963(a(n)) = 1.

A020500(a(n)) = 1. - Benoit Cloitre, Aug 26 2003

A010055(a(n)) = 0. - Reinhard Zumkeller, Nov 17 2011

a(n) ~ n. - Charles R Greathouse IV, Mar 21 2013

a(n) ~ n - pi(n) [See Panaitopol]. - N. J. A. Sloane, Sep 27 2020

A118887(a(n)) > 0. - Jonathan Sondow, Oct 17 2013