A358001 Numbers whose number of divisors is coprime to 210.

1, 1024, 4096, 59049, 65536, 262144, 531441, 4194304, 9765625, 43046721, 60466176, 241864704, 244140625, 268435456, 282475249, 387420489, 544195584, 1073741824, 2176782336, 3869835264, 10000000000, 13841287201, 15479341056, 25937424601, 31381059609, 34828517376

Offset: 1

Author

Michael De Vlieger, Dec 03 2022

Keywords

• nonn

Comments

210 is the product of the smallest 4 primes.

Numbers k such that gcd(d(k), 210) = 1, where d(k) is the number of divisors of k (A000005).

The square roots of terms are in A001694.

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..5000

Crossrefs

Subsequence of other sequences of numbers k such that gcd(d(k), m) = 1: A000290 (m=2), A336590 (m=3), A352475 (m=6), A354178 (m=30).

Cf. A000005, A001694, A008364.

Programs

Mathematica

With[{nn = 200000}, Select[Union@ Flatten@ Table[a^2*b^3, {b, nn^(1/3)}, {a, Sqrt[nn/b^3]}], CoprimeQ[DivisorSigma[0, #^2], 210] &]^2]

Formula

a(n) = A358250(n)^2.

Sum_{n>=1} 1/a(n) = Product_{p prime} (Sum_{k=2..210, gcd(k-1,210)=1} p^k)/(p^210-1) = 1.001258995976... . - Amiram Eldar, Dec 06 2022