A389776 - OEIS

%I #30 Nov 23 2025 03:49:10

%S 1296,10000,38416,46656,50625,194481,234256,456976,810000,1000000,

%T 1185921,1336336,1500625,1679616,2085136,2313441,3111696,4477456,

%U 6765201,7529536,9150625,10556001,11316496,11390625,14776336,17850625,18974736,22667121,24010000,29986576

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

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

%C A131605 is the union of A303606 and A366854.

%C A366854 is the disjoint union of this sequence and A389864.

%H Michael De Vlieger, <a href="/A389776/b389776.txt">Table of n, a(n) for n = 1..10000</a>

%F 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

%e Table of n, a(n) for n = 1..12:

%e  n       a(n)

%e -------------------------------------

%e  1      1296 =  6^4 = 2^4 * 3^4

%e  2     10000 = 10^4 = 2^4 * 5^4

%e  3     38416 = 14^4 = 2^4 * 7^4

%e  4     46656 =  6^6 = 2^6 * 3^6

%e  5     50625 = 15^4 = 3^4 * 5^4

%e  6    194481 = 21^4 = 3^4 * 7^4

%e  7    234256 = 22^4 = 2^4 * 11^4

%e  8    456976 = 26^4 = 2^4 * 13^4

%e  9    810000 = 30^4 = 2^4 * 3^4 * 5^4

%e 10   1000000 = 10^6 = 2^6 * 5^6

%e 11   1185921 = 33^4 = 3^4 * 11^4

%e 12   1336336 = 34^4 = 2^4 * 17^4

%t 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]]

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

%K nonn

%O 1,1

%A _Michael De Vlieger_, Nov 18 2025