A353415 Number of ways to write the square of n as a product of the terms of A353355 larger than 1; a(1) = 1 by convention (an empty product).

1, 1, 1, 1, 1, 3, 1, 3, 1, 2, 1, 6, 1, 3, 3, 5, 1, 6, 1, 5, 2, 2, 1, 14, 1, 3, 3, 6, 1, 12, 1, 6, 3, 2, 3, 20, 1, 3, 2, 10, 1, 12, 1, 5, 6, 2, 1, 27, 1, 5, 3, 6, 1, 14, 2, 14, 2, 3, 1, 32, 1, 2, 5, 11, 3, 12, 1, 5, 3, 12, 1, 43, 1, 3, 6, 6, 3, 12, 1, 17, 5, 2, 1, 38, 2, 3, 2, 10, 1, 38, 2, 5, 3, 2, 3, 46, 1, 6, 6

Offset: 1

Author

Antti Karttunen, Apr 19 2022

Keywords

• nonn

Comments

Number of factorizations of n^2 into factors k > 1 for which A353354(k) = 0.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..16384
• Index entries for sequences computed from indices in prime factorization

Crossrefs

Cf. A000290, A003961, A048675, A332823, A348717, A353352, A353354, A353355, A353380, A353414.

Programs

PARI

A332823(n) = { my(f = factor(n),u=(sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2)%3); if(2==u,-1,u); };
A353354(n) = sumdiv(n,d,A332823(d));
A353380(n) = (0==A353354(n));
A353414(n, m=n) = if(1==n, 1, my(s=0); fordiv(n, d, if((d>1)&&(d<=m)&&A353380(d), s += A353414(n/d, d))); (s));
A353415(n) = A353414(n^2);

Formula

a(n) = A353414(A000290(n)).

a(p) = 1 for all primes p.

a(n) = a(A003961(n)) = a(A348717(n)), for all n >= 1.