A259417 Even powers of the odd primes listed in increasing order.

1, 9, 25, 49, 81, 121, 169, 289, 361, 529, 625, 729, 841, 961, 1369, 1681, 1849, 2209, 2401, 2809, 3481, 3721, 4489, 5041, 5329, 6241, 6561, 6889, 7921, 9409, 10201, 10609, 11449, 11881, 12769, 14641, 15625, 16129, 17161, 18769, 19321, 22201, 22801, 24649

Offset

1,2

Comments

Each of the following sequences, p^(q-1) with p >= 2 and q > 2 primes, except their respective first elements, powers of 2, is a subsequence:

A001248(p) = p^2, A030514(p) = p^4, A030516(p) = p^6,

A030629(p) = p^10, A030631(p) = p^12, A030635(p) = p^16,

A030637(p) = p^18, A137486(p) = p^22, A137492(p) = p^28,

A139571(p) = p^30, A139572(p) = p^36, A139573(p) = p^40,

A139574(p) = p^42, A139575(p) = p^46, A173533(p) = p^52,

A183062(p) = p^58, A183085(p) = p^60.

See also the link to the OEIS Wiki.

The sequences A053182(n)^2, A065509(n)^4, A163268(n)^6 and A240693(n)^10 are subsequences of this sequence.

The odd numbers in A023194 are a subsequence of this sequence.

Links

Hartmut F. W. Hoft, Table of n, a(n) for n = 1..473

OEIS Wiki, Index entries for number of divisors.

Formula

Sum_{n>=1} 1/a(n) = 1 + Sum_{k>=1} (P(2*k) - 1/2^(2*k)) = 1.21835996432366585110..., where P is the prime zeta function. - Amiram Eldar, Jul 10 2022

Example

a(11) = 5^4 = 625 is followed by a(12) = 3^6 = 729 since no even power of an odd prime falls between them.

Mathematica

a259417[bound_] := Module[{q, h, column = {}}, For[q = Prime[2], q^2 <= bound, q = NextPrime[q], For[h = 1, q^(2*h) <= bound, h++, AppendTo[column, q^(2*h)]]]; Prepend[Sort[column], 1]]
a259417[25000] (* data *)
With[{upto=25000}, Select[Union[Flatten[Table[Prime[Range[2, Floor[ Sqrt[ upto]]]]^n, {n, 0, Log[2, upto], 2}]]], #<=upto&]] (* Harvey P. Dale, Nov 25 2017 *)

Crossrefs

Sequence in context: A348742 A016754 A110487 * A030156 A331587 A192775

Adjacent sequences: A259414 A259415 A259416 * A259418 A259419 A259420

Keyword

nonn

Author

Hartmut F. W. Hoft, Jun 26 2015

Status

approved