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

1,6

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

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.

Prog

(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);

Crossrefs

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

Sequence in context: A331286 A272880 A003636 * A078929 A030728 A237712

Adjacent sequences: A353412 A353413 A353414 * A353416 A353417 A353418

Keyword

nonn

Author

Antti Karttunen, Apr 19 2022

Status

approved