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

Author

Hartmut F. W. Hoft, Jun 26 2015

Keywords

• nonn

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.

Examples

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

Links

• Hartmut F. W. Hoft, Table of n, a(n) for n = 1..473
• OEIS Wiki, Index entries for number of divisors.

Programs

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 *)

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