OFFSET
1,2
COMMENTS
All terms are square since numbers coprime to 6 are odd.
The square roots of terms are in A001694.
Intersection of A000290 and A336590, i.e., numbers whose prime factorization has only exponents that are congruent to {0, 4} mod 6 (A047233). - Amiram Eldar, Mar 31 2022
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..10000
Titus Hilberdink, How often is d(n) a power of a given integer?, Journal of Number Theory, Vol. 236 (2022), pp. 261-279.
FORMULA
a(n) = A350014(n)^2.
Sum_{n>=1} 1/a(n) = Pi^2/9 (A100044). - Amiram Eldar, Mar 31 2022
The number of terms <= x is (zeta(3/2)/zeta(2))*x^(1/4) + (zeta(2/3)/zeta(4/3))*x^(1/6) + O(x^(1/8 + eps)), for all eps > 0 (Hilberdink, 2022). - Amiram Eldar, May 18 2022
MATHEMATICA
Select[Range[864]^2, GCD[DivisorSigma[0, #], 6] == 1 &] (* or, more efficiently, *)
With[{nn = 864}, Select[Union[Flatten@ Table[a^2*b^3, {b, nn^(1/3)}, {a, Sqrt[nn/b^3]}]]^2, Mod[DivisorSigma[0, #], 3] > 0 &]]
PROG
(PARI) isok(m) = gcd(numdiv(m), 6) == 1; \\ Michel Marcus, Mar 29 2022
(PARI) m = 100000; seq = direuler(p=2, m, (1 - X^8)/(1 - X^4)/(1 - X^6)); for(n=1, m, if(seq[n] != 0, print1(n, ", "))) \\ Vaclav Kotesovec, May 19 2022
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Michael De Vlieger, Mar 26 2022
STATUS
approved