A366243 - OEIS

%I #7 Oct 06 2023 10:56:02

%S 1,4,9,25,36,49,100,121,169,196,225,256,289,361,441,484,529,676,841,

%T 900,961,1024,1089,1156,1225,1369,1444,1521,1681,1764,1849,2116,2209,

%U 2304,2601,2809,3025,3249,3364,3481,3721,3844,4225,4356,4489,4761,4900,5041,5329

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

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

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

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

%C 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).

%H Amiram Eldar, <a href="/A366243/b366243.txt">Table of n, a(n) for n = 1..10000</a>

%F 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).

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

%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(e[i]%2 || !ismd(e[i]/2), return(0))); 1;}

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

%Y A062503 is a subsequence.

%K nonn,easy

%O 1,2

%A _Amiram Eldar_, Oct 05 2023