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

1,1

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

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

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).

Formula

A001221(a(n)) > 1.

A014963(a(n)) = 1.

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

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

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 *)

Prog

(Magma) IsA024619:=func< n | not IsPrime(n) and not (t and IsPrime(b) where t, b, _:=IsPower(n)) >; [ n: n in [2..200] | IsA024619(n) ]; // Klaus Brockhaus, Feb 25 2011
(Haskell)
a024619 n = a024619_list !! (n-1)
a024619_list = filter ((== 0) . a010055) [1..]
-- Reinhard Zumkeller, Nov 17 2011
(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]
(PARI) is(n)=n>5 && !isprimepower(n) \\ Charles R Greathouse IV, Mar 21 2013

Crossrefs

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

Cf. A340511.

Subsequence of A080257.

Sequence in context: A105642 A064040 A168638 * A361102 A335080 A323304

Adjacent sequences: A024616 A024617 A024618 * A024620 A024621 A024622

Keyword

nonn,easy

Author

Clark Kimberling

Status

approved