A354178 Numbers whose number of divisors is coprime to 30.

1, 64, 729, 1024, 4096, 15625, 46656, 59049, 65536, 117649, 262144, 531441, 746496, 1000000, 1771561, 2985984, 3779136, 4194304, 4826809, 7529536, 9765625, 11390625, 16000000, 24137569, 34012224, 43046721, 47045881, 47775744, 60466176, 64000000, 85766121, 113379904

Offset

1,2

Comments

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

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

Links

Michael De Vlieger, Table of n, a(n) for n = 1..10000 (first 1709 terms from Amiram Eldar)

Titus Hilberdink, How often is d(n) a power of a given integer?, Journal of Number Theory, Vol. 236 (2022), pp. 261-279.

Index entries for sequences computed from exponents in factorization of n.

Formula

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

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).

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...

Example

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

Mathematica

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

Prog

(PARI) isok(k) = gcd(numdiv(k), 30) == 1;
for(k=1, 10650, if(isok(k^2), print1(k^2, ", ")))

Crossrefs

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

Cf. A000005, A007775, A354179.

Sequence in context: A074471 A164345 A164337 * A367803 A161860 A195249

Adjacent sequences: A354175 A354176 A354177 * A354179 A354180 A354181

Keyword

nonn

Author

Amiram Eldar, May 18 2022

Status

approved