A216419 Odd powers that are not prime powers.

225, 441, 1089, 1225, 1521, 2025, 2601, 3025, 3249, 3375, 3969, 4225, 4761, 5625, 5929, 7225, 7569, 8281, 8649, 9025, 9261, 9801, 11025, 12321, 13225, 13689, 14161, 15129, 16641, 17689, 18225, 19881, 20449, 21025, 21609, 23409, 24025, 25281, 25921, 27225

Offset

1,1

Comments

Numbers in A075109 but not in A000961.

Also odd perfect powers having no primitive root (intersection of A075109 and A175594).

Links

Arkadiusz Wesolowski, Table of n, a(n) for n = 1..10000

Eric Weisstein's World of Mathematics, Primitive Root.

Formula

Sum_{n>=1} 1/a(n) = 1/2 + Sum_{k>=2} mu(k)*(1-zeta(k)*(2^k-1)/2^k) - Sum_{p prime} 1/(p*(p-1)) = 0.0158808884... - Amiram Eldar, Dec 21 2020

Example

81 = 9^2 as well as 81 = 3^4, therefore 81 is not a term.
225 can be expressed so in one way as (3*5)^2, therefore 225 is a term.

Mathematica

nn = 27500; lst = Union[Flatten[Table[n^i, {i, Prime[Range[PrimePi[Log[2, nn]]]]}, {n, 2, nn^(1/i)}]]]; Select[lst, OddQ[#] && ! IntegerQ@PrimitiveRoot[#] &] (* Most of the code is from T. D. Noe *)

Prog

(Magma) [n : n in [3..27225 by 2] | IsPower(n) and EulerPhi(n) ne CarmichaelLambda(n)]; // Arkadiusz Wesolowski, Nov 09 2013

Crossrefs

Cf. A008683, A075109, A136141, A175594.

Sequence in context: A267892 A363217 A373087 * A246199 A147276 A219022

Adjacent sequences: A216416 A216417 A216418 * A216420 A216421 A216422

Keyword

nonn

Author

Arkadiusz Wesolowski, Sep 06 2012

Status

approved