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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.
1
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
1,2
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)
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.
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]
PROG
(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; }
CROSSREFS
Intersection of A001694 and A366242.
Intersection of A000583 and A366242.
A113849 is a subsequence.
Sequence in context: A377022 A153157 A369168 * A113849 A046453 A030514
KEYWORD
nonn,easy
AUTHOR
Amiram Eldar, Oct 06 2023
STATUS
approved