A389776 Powers k^m, even m >= 4, where k is composite and squarefree.

1296, 10000, 38416, 46656, 50625, 194481, 234256, 456976, 810000, 1000000, 1185921, 1336336, 1500625, 1679616, 2085136, 2313441, 3111696, 4477456, 6765201, 7529536, 9150625, 10556001, 11316496, 11390625, 14776336, 17850625, 18974736, 22667121, 24010000, 29986576

Offset

1,1

Comments

Intersection of A303606 (perfect powers of squarefree composites) and A366854 (perfect powers of numbers that are neither squarefree nor prime powers).

A131605 is the union of A303606 and A366854.

A366854 is the disjoint union of this sequence and A389864.

Links

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Formula

Sum_{n>=1} 1/a(n) = Sum_{k>=2} (zeta(2*k)/zeta(4*k)- P(2*k) - 1) = 0.00095830286979623662479..., where P is the prime zeta function. - Amiram Eldar, Nov 23 2025

Example

Table of n, a(n) for n = 1..12:
 n       a(n)
-------------------------------------
 1      1296 =  6^4 = 2^4 * 3^4
 2     10000 = 10^4 = 2^4 * 5^4
 3     38416 = 14^4 = 2^4 * 7^4
 4     46656 =  6^6 = 2^6 * 3^6
 5     50625 = 15^4 = 3^4 * 5^4
 6    194481 = 21^4 = 3^4 * 7^4
 7    234256 = 22^4 = 2^4 * 11^4
 8    456976 = 26^4 = 2^4 * 13^4
 9    810000 = 30^4 = 2^4 * 3^4 * 5^4
10   1000000 = 10^6 = 2^6 * 5^6
11   1185921 = 33^4 = 3^4 * 11^4
12   1336336 = 34^4 = 2^4 * 17^4

Mathematica

nn = 3 * 10^7; k = 4; m = 2; mm = Surd[nn, m]; i = 1; MapIndexed[Set[S[First[#2]], #1] &, Select[Range[mm], And[SquareFreeQ[#], CompositeQ[#] ] &] ]; Union@ Reap[While[j = k; While[S[i]^j < nn, Sow[S[i]^j]; j += m]; j > k, i++] ][[-1, 1]]

Crossrefs

Cf. A001597, A001694, A036967, A120944, A126706, A131605, A286708, A303606, A366854, A389864.

Sequence in context: A320893 A390127 A276282 * A137488 A189991 A250438

Adjacent sequences: A389773 A389774 A389775 * A389777 A389778 A389779

Keyword

nonn

Author

Michael De Vlieger, Nov 18 2025

Status

approved