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A366307 Powerful numbers that are products of "Fermi-Dirac primes" (A050376) that are powers of primes with exponents that are powers of 4.

%I A366307 #9 Oct 07 2023 08:48:08
%S A366307 1,16,81,625,1296,2401,10000,14641,28561,38416,50625,65536,83521,
%T A366307 130321,194481,234256,279841,456976,707281,810000,923521,1048576,
%U A366307 1185921,1336336,1500625,1874161,2085136,2313441,2825761,3111696,3418801,4477456,4879681,5308416,6765201
%N A366307 Powerful numbers that are products of "Fermi-Dirac primes" (A050376) that are powers of primes with exponents that are powers of 4.
%C A366307 Equivalently, powerful numbers that are products of "Fermi-Dirac primes" that are powers of primes with exponents that are powers of 2 with even exponents.
%C A366307 Products of distinct numbers of the form p^(2^(2*k)), where p is prime and k >= 1.
%C A366307 Numbers whose prime factorization has exponents that are the even positive terms of the Moser-de Bruijn sequence (A000695).
%H A366307 Amiram Eldar, <a href="/A366307/b366307.txt">Table of n, a(n) for n = 1..11659</a> (terms below 10^17)
%F A366307 a(n) = A366242(n)^4 = A000583(A366242(n)).
%F A366307 Sum_{n>=1} 1/a(n) = Product_{k>=1} zeta(2^(2*k))/zeta(2^(2*k+1)) = 1.07794460966828564964... = zeta(2)/c, where c is the constant defined in A366242.
%t A366307 mdQ[n_] := AllTrue[IntegerDigits[n, 4], # < 2 &]; Select[Range[10^6], # == 1 || AllTrue[FactorInteger[#][[;; , 2]], EvenQ[#1] && mdQ[#1] &] &]
%t A366307 (* or *)
%t A366307 seq[max_] := Module[{ps = {2}, p, s = {1}, s1, s2, emax}, While[ps[[-1]]^4 < max, AppendTo[ps, NextPrime[ps[[-1]]]]]; Do[p = ps[[k]]; emax = Floor[Log2[Floor[Log[p, max]]]]; Do[s1 = {1, p^(2^e)}; s2 = Select[Union[Flatten[Outer[Times, s, s1]]], # <= max &]; s = Union[s, s2], {e, 2, emax, 2}], {k, 1, Length[ps]}]; s]; seq[10^7]
%o A366307 (PARI) ismd(n) = {my(d = digits(n, 4)); for(i = 1, #d, if(d[i] > 1, return(0))); 1;}
%o A366307 is(n) = {my(e = factor(n)[,2]); for(i = 1, #e, if(e[i]%2 || !ismd(e[i]), return(0))); 1;}
%Y A366307 Cf. A000695, A013661, A050376.
%Y A366307 Intersection of A001694 and A366242.
%Y A366307 Intersection of A000583 and A366242.
%Y A366307 A113849 is a subsequence.
%K A366307 nonn,easy
%O A366307 1,2
%A A366307 _Amiram Eldar_, Oct 06 2023