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

Author

Amiram Eldar, Oct 05 2023

Keywords

• nonn
• easy

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

Crossrefs

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

A062503 is a subsequence.

Programs

Mathematica

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

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;}

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).