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Numbers that are products of "Fermi-Dirac primes" (A050376) that are powers of primes with exponents that are powers of 4.
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%I #8 Oct 06 2023 10:55:29

%S 1,2,3,5,6,7,10,11,13,14,15,16,17,19,21,22,23,26,29,30,31,32,33,34,35,

%T 37,38,39,41,42,43,46,47,48,51,53,55,57,58,59,61,62,65,66,67,69,70,71,

%U 73,74,77,78,79,80,81,82,83,85,86,87,89,91,93,94,95,96,97

%N Numbers that are products of "Fermi-Dirac primes" (A050376) that are powers of primes with exponents that are powers of 4.

%C A subsequence of A252895, and first differs from it at n = 172. A252895(172) = 256 = 2^(2^3) is not a term of this sequence.

%C Equivalently, numbers that are products of "Fermi-Dirac primes" that are powers of primes with exponents that are powers of 2 with even exponents.

%C Products of distinct numbers of the form p^(2^(2*k)), where p is prime and k >= 0.

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

%C Every integer k has a unique representation as a product of 2 numbers: one is in this sequence and the other is in A366243: k = A366244(k) * A366245(k).

%C The asymptotic density of this sequence is 1/c = 0.65531174251481086750..., where c is given in the formula section.

%H Amiram Eldar, <a href="/A366242/b366242.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) ~ c * n, where c = Product_{k>=0} zeta(2^(2*k+1))/zeta(2^(2*k+2)) = 1.52599127273749217982... .

%t mdQ[n_] := AllTrue[IntegerDigits[n, 4], # < 2 &]; Select[Range[100], AllTrue[FactorInteger[#][[;; , 2]], mdQ] &]

%o (PARI) ismd(n) = {my(d = digits(n, 4)); for(i = 1, #d, if(d[i] > 1, return(0))); 1;}

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

%Y Cf. A000695, A050376, A366243, A366244, A366245.

%Y Subsequence of A252895.

%K nonn,easy

%O 1,2

%A _Amiram Eldar_, Oct 05 2023