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

1,2

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

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

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] &]]

Prog

(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

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.

Sequence in context: A069492 A076469 A256819 * A235057 A339358 A249116

Adjacent sequences: A358247 A358248 A358249 * A358251 A358252 A358253

Keyword

nonn

Author

Michael De Vlieger, Dec 03 2022

Status

approved