A366243 Numbers that are products of "Fermi-Dirac primes" (A050376) that are powers of primes with exponents that are not powers of 4.

1, 4, 9, 25, 36, 49, 100, 121, 169, 196, 225, 256, 289, 361, 441, 484, 529, 676, 841, 900, 961, 1024, 1089, 1156, 1225, 1369, 1444, 1521, 1681, 1764, 1849, 2116, 2209, 2304, 2601, 2809, 3025, 3249, 3364, 3481, 3721, 3844, 4225, 4356, 4489, 4761, 4900, 5041, 5329

Offset

1,2

Comments

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

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

Numbers whose prime factorization has exponents that are positive terms of A062880.

Every integer k has a unique representation as a product of 2 numbers: one is in this sequence and the other is in A366242: k = A366245(k) * A366244(k).

Links

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

Formula

Sum_{n>=1} 1/a(n) = Product_{k>=0} zeta(2^(2*k+1))/zeta(2^(2*k+2)) = 1.52599127273749217982... (this is the constant c in A366242).

Mathematica

mdQ[n_] := AllTrue[IntegerDigits[n, 4], # < 2 &]; q[e_] := EvenQ[e] && mdQ[e/2]; Select[Range[6000], # == 1 || AllTrue[FactorInteger[#][[;; , 2]], q] &]

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]/2), return(0))); 1; }

Crossrefs

Cf. A062880, A050376, A366242, A366244, A366245.

A062503 is a subsequence.

Sequence in context: A355058 A153158 A111245 * A062503 A248648 A063577

Adjacent sequences: A366240 A366241 A366242 * A366244 A366245 A366246

Keyword

nonn,easy

Author

Amiram Eldar, Oct 05 2023

Status

approved