A366242 Numbers that are products of "Fermi-Dirac primes" (A050376) that are powers of primes with exponents that are powers of 4.

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, 37, 38, 39, 41, 42, 43, 46, 47, 48, 51, 53, 55, 57, 58, 59, 61, 62, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 80, 81, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 96, 97

Offset

1,2

Comments

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.

Equivalently, 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 >= 0.

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

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).

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

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Formula

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

Mathematica

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

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(!ismd(e[i]), return(0))); 1; }

Crossrefs

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

Subsequence of A252895.

Sequence in context: A059266 A336222 A252895 * A336224 A274034 A197680

Adjacent sequences: A366239 A366240 A366241 * A366243 A366244 A366245

Keyword

nonn,easy

Author

Amiram Eldar, Oct 05 2023

Status

approved