This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A372405 #39 May 12 2024 02:01:12
%S A372405 1,16,81,256,512,625,1296,2401,6561,10000,14641,19683,20736,28561,
%T A372405 38416,41472,50625,65536,83521,104976,130321,160000,194481,234256,
%U A372405 279841,314928,320000,390625,456976,614656,707281,810000,923521,1185921,1229312,1336336,1500625,1679616
%N A372405 Exponentially powerful numbers whose prime factorization exponents are all powerful numbers > 1.
%C A372405 In other words, numbers m such that if p^k is the greatest power of any prime p which divides m, then k is a term > 1 in A001694.
%C A372405 Subsequence of A001694 (since all prime exponents are > 1).
%C A372405 Compare with A361177, of which this is a subsequence (see Formula).
%C A372405 Distinct from A277562; A277652(26) = 331776 = 2^12 * 3^4 is not in this sequence. - _Michael De Vlieger_, Apr 30 2024
%C A372405 1 and 41472 are two terms here that are not in A277562. - _David A. Corneth_, Apr 30 2024
%H A372405 David A. Corneth, <a href="/A372405/b372405.txt">Table of n, a(n) for n = 1..10000</a> (first 1580 terms from Michael De Vlieger)
%F A372405 Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + Sum_{k>=2} 1/p^A001694(k)) = 1.08410926642148594327... . - _Amiram Eldar_, May 12 2024
%e A372405 16 = 2^4 and 4 = A001694(2) is a powerful number.
%e A372405 a(7) = 1296 = 2^4*3^4.
%e A372405 a(12) = 19683 = 3^9 (9 = A001694(4) is a powerful number).
%t A372405 nn = 2^21; f[x_] := f[x] = Times @@ FactorInteger[x][[All, 1]]; Select[Union@ Flatten@ Table[a^7*b^6*c^5*d^4, {d, Surd[nn, 4]}, {c, Surd[nn/d^4, 5]}, {b, Surd[nn/(c^5*d^4), 6]}, {a, Surd[nn/(b^6*c^5*d^4), 7]}], AllTrue[FactorInteger[#][[All, -1]], Divisible[#, f[#]^2] &] &] (* _Michael De Vlieger_, Apr 29 2024 *)
%o A372405 (PARI) isok(k) = if (ispowerful(k), my(f=factor(k)[,2]); #select(ispowerful, f) == #f); \\ _Michel Marcus_, Apr 30 2024
%Y A372405 Intersection of A001694 and A361177.
%Y A372405 Subsequence of A036967.
%K A372405 nonn
%O A372405 1,2
%A A372405 _David James Sycamore_, Apr 29 2024
%E A372405 More terms from _Michael De Vlieger_, Apr 29 2024