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

1, 16, 81, 625, 1296, 2401, 10000, 14641, 28561, 38416, 50625, 65536, 83521, 130321, 194481, 234256, 279841, 456976, 707281, 810000, 923521, 1048576, 1185921, 1336336, 1500625, 1874161, 2085136, 2313441, 2825761, 3111696, 3418801, 4477456, 4879681, 5308416, 6765201

Offset: 1

Author

Amiram Eldar, Oct 06 2023

Keywords

• nonn
• easy

Comments

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.

Products of distinct numbers of the form p^(2^(2*k)), where p is prime and k >= 1.

Numbers whose prime factorization has exponents that are the even positive terms of the Moser-de Bruijn sequence (A000695).

Links

• Amiram Eldar, Table of n, a(n) for n = 1..11659 (terms below 10^17)

Crossrefs

Cf. A000695, A013661, A050376.

Intersection of A001694 and A366242.

Intersection of A000583 and A366242.

A113849 is a subsequence.

Programs

Mathematica

mdQ[n_] := AllTrue[IntegerDigits[n, 4], # < 2 &]; Select[Range[10^6], # == 1 || AllTrue[FactorInteger[#][[;; , 2]], EvenQ[#1] && mdQ[#1] &] &]
(* or *)
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]

PARI

ismd(n) = {my(d = digits(n, 4)); for(i = 1, #d, if(d[i] > 1, return(0))); 1;}
is(n) = {my(e = factor(n)[,2]); for(i = 1, #e, if(e[i]%2 || !ismd(e[i]), return(0))); 1;}

Formula

a(n) = A366242(n)^4 = A000583(A366242(n)).

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.