A358250 Numbers whose square has a number of divisors coprime to 210.

1, 32, 64, 243, 256, 512, 729, 2048, 3125, 6561, 7776, 15552, 15625, 16384, 16807, 19683, 23328, 32768, 46656, 62208, 100000, 117649, 124416, 161051, 177147, 186624, 200000, 209952, 262144, 371293, 373248, 390625, 419904, 497664, 500000, 537824, 629856, 759375

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^2), 210) = 1, where d(k) is the number of divisors of k (A000005).

Also numbers with no exponents = 1 mod 3, 2 mod 5, or 3 mod 7; also numbers whose square has a number of divisors coprime to 105. - Charles R Greathouse IV, Dec 08 2022

Links

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

Crossrefs

Subsequence of A069492 and hence of A036967, A036966, and A001694.

Subsequence of other sequences of numbers k such that gcd(d(k^2), m) = 1: A350014 (m=6), A354179 (m=30).

Cf. A000005, A000290, A008364.

Programs

Mathematica

With[{nn = 2^20}, Select[Union@ Flatten@ Table[a^2*b^3, {b, nn^(1/3)}, {a, Sqrt[nn/b^3]}], CoprimeQ[DivisorSigma[0, #^2], 210] &]]

PARI

is(n,f=factor(n))=if(n<32, return(n==1)); my(t=f[,2]%105, N=19200959813818273241621521446046); for(i=1,#t, if(bittest(N,t[i]), return(0))); 1 \\ Charles R Greathouse IV, Dec 08 2022

Formula

Sum_{n>=1} 1/a(n) = Product_{p prime} (Sum_{k=2..210, gcd(k-1,210)=1} p^(k/2))/(p^105-1) = 1.05981355805... . - Amiram Eldar, Dec 06 2022