OFFSET
1,2
COMMENTS
For each term k, let p^e be the highest power of a prime p that divides k. Then e >= 4 is even for p == 3 (mod 4), or e >= 3 otherwise.
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..10000
Rafael Jakimczuk, Generalizations of Mertens's Formula and k-Free and s-Full Numbers with Prime Divisors in Arithmetic Progression, ResearchGate, 2024.
FORMULA
The number of terms that do not exceed x is ~ c * x^(1/3)/sqrt(log(x)), where c = (6/Pi^2) * sqrt(3) * (1 + 1/(3*(2^(1/3)-1))) * B * Product_{primes p == 1 (mod 4)} (1 + 1/((p^(1/3)-1)*(p+1))) * Product_{primes p == 3 (mod 4)} (1 + p^(1/3)/(p*(p^(2/3)-1))), where B is the Landau-Ramanujan constant (A064533) (Jakimczuk, 2024, Theorem 4.5, p. 47).
Sum_{n>=1} 1/a(n) = (5/4) * Product_{primes p == 1 (mod 4)} (1 + 1/(p^2*(p-1))) * Product_{primes p == 3 (mod 4)} (1 + 1/(p^2*(p^2-1))) = 1.281719491797642498... .
MATHEMATICA
Select[Range[25000], SquaresR[2, #] > 0 && (# == 1 || Min[FactorInteger[#][[;; , 2]]] > 2) &]
PROG
(PARI) is(n) = {my(f=factor(n)); for(i=1, #f~, if(f[i, 2] < 3 || (f[i, 2]%2 && f[i, 1]%4 == 3), return(0))); 1; }
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Amiram Eldar, Mar 08 2024
STATUS
approved