

A024619


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


63



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
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OFFSET

1,1


COMMENTS

Since 1 = p^0 does not have a well defined prime base p it is often excluded from the prime powers, in which case 1 would be prepended to this sequence to give the complement of "Prime powers p^k (k >= 1)".  Daniel Forgues, Mar 02 2009
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 ngon 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


LINKS

Daniel Forgues and Reinhard Zumkeller, Table of n, a(n) for n = 1..10000 (first 8719 terms from Daniel Forgues)
Marius Munteanu and Laura Munteanu, Rational equiangular polygons, Applied Math., 4 (2013), 14601465.
Eric Weisstein's World of Mathematics, Prime Power
Wikipedia, Prime power
G. J. Woeginger, Nothing new about equiangular polygons, Amer. Math. Monthly, 120 (2013), 849850.
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
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 !! (n1)
a024619_list = filter ((== 0) . a010055) [1..]
 Reinhard Zumkeller, Nov 17 2011
(Sage)
def A024619_list(n) :
return [k for k in (1..n) if not k.is_prime() and not k.is_perfect_power()]
A024619_list(112) # Peter Luschny, Feb 03 2012
(PARI) is(n)=n>5 && !isprimepower(n) \\ Charles R Greathouse IV, Mar 21 2013


CROSSREFS

Cf. A000040, A000961, A001221, A014963, A020500, A085970.
Subsequence of A080257.
Sequence in context: A105642 A064040 A168638 * A106543 A007774 A030231
Adjacent sequences: A024616 A024617 A024618 * A024620 A024621 A024622


KEYWORD

nonn,easy


AUTHOR

Clark Kimberling


STATUS

approved



