A358250 - OEIS

%I #14 Dec 08 2022 09:55:02

%S 1,32,64,243,256,512,729,2048,3125,6561,7776,15552,15625,16384,16807,

%T 19683,23328,32768,46656,62208,100000,117649,124416,161051,177147,

%U 186624,200000,209952,262144,371293,373248,390625,419904,497664,500000,537824,629856,759375

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

%C 210 is the product of the smallest 4 primes.

%C Numbers k such that gcd(d(k^2), 210) = 1, where d(k) is the number of divisors of k (A000005).

%C 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

%H Michael De Vlieger, <a href="/A358250/b358250.txt">Table of n, a(n) for n = 1..5000</a>

%F 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

%t 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] &]]

%o (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

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

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

%Y Cf. A000005, A000290, A008364.

%K nonn

%O 1,2

%A _Michael De Vlieger_, Dec 03 2022