A354178 - OEIS

%I #22 Oct 27 2023 19:31:40

%S 1,64,729,1024,4096,15625,46656,59049,65536,117649,262144,531441,

%T 746496,1000000,1771561,2985984,3779136,4194304,4826809,7529536,

%U 9765625,11390625,16000000,24137569,34012224,43046721,47045881,47775744,60466176,64000000,85766121,113379904

%N Numbers whose number of divisors is coprime to 30.

%C Numbers k such that gcd(d(k), 30) = 1, where d(k) is the number of divisors of k (A000005).

%C All the terms are squares since their number of divisors is odd.

%H Michael De Vlieger, <a href="/A354178/b354178.txt">Table of n, a(n) for n = 1..10000</a> (first 1709 terms from Amiram Eldar)

%H Titus Hilberdink, <a href="https://doi.org/10.1016/j.jnt.2021.07.020">How often is d(n) a power of a given integer?</a>, Journal of Number Theory, Vol. 236 (2022), pp. 261-279.

%H <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>.

%F a(n) = A354179(n)^2.

%F The number of terms <= x is (zeta(5)*zeta(5/3))/(zeta(4)*zeta(10/3))*x^(1/6) + (zeta(3)*zeta(3/5))/(zeta(2)*zeta(12/5))*x^(1/10) + O(x^(1/20 + eps)) for all eps > 0 (Hilberdink, 2022).

%F Sum_{n>=1} 1/a(n) = Product_{p prime} (p^2 + p^8 + p^12 + p^14 + p^18 + p^20 + p^24 + p^30)/(p^30 - 1) = 1.0183538548...

%e 64 is a term since A000005(64) = 7 and gcd(7, 30) = 1.

%t Select[Range[10^4]^2, CoprimeQ[DivisorSigma[0, #], 30] &]

%o (PARI) isok(k) = gcd(numdiv(k), 30) == 1;

%o for(k=1, 10650, if(isok(k^2), print1(k^2,", ")))

%Y Subsequence of other sequences of numbers k such that gcd(d(k), m) = 1: A000290 (m=2), A336590 (m=3), A352475 (m=6).

%Y Cf. A000005, A007775, A354179.

%K nonn

%O 1,2

%A _Amiram Eldar_, May 18 2022