OFFSET
1,2
COMMENTS
The asymptotic density of this sequence is Product_{primes p} (1 - 1/p^2 + 1/p^3) = 0.748535... (A330596).
A304364 is apparently a subsequence.
These numbers were named semi-2-free integers by Suryanarayana (1971). - Amiram Eldar, Dec 29 2020
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..10000
D. Suryanarayana, Semi-k-free integers, Elemente der Mathematik, Vol. 26 (1971), pp. 39-40.
D. Suryanarayana and R. Sitaramachandra Rao, Distribution of semi-k-free integers, Proceedings of the American Mathematical Society, Vol. 37, No. 2 (1973), pp. 340-346.
FORMULA
Sum_{n>=1} 1/a(n)^s = zeta(s) * Product_{p prime} (1 - 1/p^(2*s) + 1/p^(3*s)), for s > 1. - Amiram Eldar, Oct 21 2023
EXAMPLE
6 = 2^1 * 3^1 is a term since none of the exponents in its prime factorization is equal to 2.
9 = 3^2 is not a term since it has an exponent 2 in its prime factorization.
MAPLE
q:= n-> andmap(i-> i[2]<>2, ifactors(n)[2]):
select(q, [$1..100])[]; # Alois P. Heinz, Aug 12 2020
MATHEMATICA
Select[Range[100], !MemberQ[FactorInteger[#][[;; , 2]], 2] &]
PROG
(PARI) is(n) = {my(f = factor(n)); for(i = 1, #f~, if(f[i, 2] == 2, return(0))); 1; } \\ Amiram Eldar, Oct 21 2023
CROSSREFS
Complement of A038109.
KEYWORD
nonn,easy
AUTHOR
Amiram Eldar, Aug 12 2020
STATUS
approved