A051676 -id:A051676 - cp's OEIS frontend

A350014 Numbers whose square has a number of divisors coprime to 6.

1, 4, 8, 9, 25, 27, 32, 36, 49, 64, 72, 100, 108, 121, 125, 169, 196, 200, 216, 225, 243, 256, 288, 289, 343, 361, 392, 441, 484, 500, 512, 529, 576, 675, 676, 729, 800, 841, 864, 900, 961, 968, 972, 1000, 1089, 1125, 1156, 1225, 1323, 1331, 1352, 1369, 1372, 1444

Offset: 1

Michael De Vlieger, Jan 17 2022

a(n) = m in A001694 such that d(m^2) is not divisible by 3, where d(n) = A000005(n).
Supersequence of A051676 (composite numbers whose square has a prime number of divisors).
Subsequence of A001694 (powerful numbers).
Numbers whose prime factorization has only exponents that are congruent to {0, 2} mod 3 (A007494). - Amiram Eldar, Mar 31 2022

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A000005, A000290, A007494, A001694, A001651, A051676, A240976.

A350014 := proc(n)
    option remember ;
    local a;
    if n =1 then
        1;
    else
        for a from procname(n-1)+1 do
            if igcd(numtheory[tau](a^2),6) = 1 then
                return a;
            end if;
        end do:
    end if;
end proc:
seq(A350014(n),n=1..20) ; # R. J. Mathar, Apr 06 2022

Select[Range[1500], CoprimeQ[DivisorSigma[0, #^2], 6] &] (* or *)
With[{nn = 1500}, Select[Union@ Flatten@ Table[a^2*b^3, {b, nn^(1/3)}, {a, Sqrt[nn/b^3]}], Mod[DivisorSigma[0, #^2], 3] != 0 &]]

isok(m) = gcd(numdiv(m^2), 6) == 1; \\ Michel Marcus, Mar 04 2022

a(n) = {m : gcd(d(m^2), 6) = 1}.

Sum_{n>=1} 1/a(n) = 15*zeta(3)/Pi^2 (= 10 * A240976). - Amiram Eldar, Mar 31 2022

A283262 Numbers m such that tau(m^2) is a prime.

Original entry on oeis.org

2, 3, 4, 5, 7, 8, 9, 11, 13, 17, 19, 23, 25, 27, 29, 31, 32, 37, 41, 43, 47, 49, 53, 59, 61, 64, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 121, 125, 127, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227

Offset: 1

Jaroslav Krizek, Mar 08 2017

nonn

tau(m) is the number of positive divisors of m (A000005).
Numbers m such that A000005(A000290(m)) = A048691(m) is a prime.
Union of A000040 (primes) and A051676.
Supersequence of A055638 (sigma(m^2) is prime).
Subsequence of A000961 (powers of primes).
Prime powers p^e with 2e+1 prime (e >= 1).
See A061285(m) = the smallest number k such that tau(k^2) = m-th prime.

			tau(4^2) = tau(16) = 5 (prime).

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000005, A000290, A000961, A048691, A051676, A055638, A061285.

[n: n in [2..100000] | IsPrime(NumberOfDivisors(n^2))];

N:= 1000: # to get all terms <= N
es:= select(t -> isprime(2*t+1), [$1..ilog2(N)]):
Ps:= select(isprime, [2,seq(i,i=3..N,2)]):
sort(select(`<=`, [seq(seq(p^e,e=es),p=Ps)],N)): # Robert Israel, Mar 16 2017

Select[Range@ 227, PrimeQ[DivisorSigma[0, #^2]] &] (* Michael De Vlieger, Mar 09 2017 *)

isok(n)=isprime(numdiv(n^2)) \\ Indranil Ghosh, Mar 09 2017

cp's OEIS Frontend

A350014 Numbers whose square has a number of divisors coprime to 6.

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