A353303 Number of factorizations of n into factors k > 1 for which A156552(k) is a multiple of three.

1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 0, 1, 2, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 3, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 1, 0, 2, 1, 0, 1, 2, 0, 2, 1, 0, 0, 1, 0, 0, 0, 0, 0, 3, 0, 1, 0, 0, 1

Offset: 1

Author

Antti Karttunen, Apr 10 2022

Keywords

• nonn

Comments

Number of factorizations of n into terms of A329609 that are larger than one.

Examples

			Divisors of 16 are [1, 2, 4, 8, 16]. When we apply A156552 to them, we obtain [0, 1, 3, 7, 15], of which only 0, 3 and 15 are multiples of three, therefore only factorizations 1*16 and 4*4 of 16 are counted, therefore a(16) = 2.
792 has 24 divisors in total, but only d = [1, 4, 9, 22, 36, 66, 88, 198, 264, 792] are such that A156552(d) is a multiple of 3. When using them, the following five factorizations are possible: 792 = 4*198 = 9*88 = 22*36 = 4*9*22, therefore a(792) = 5.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537
• Index entries for sequences computed from indices in prime factorization

Crossrefs

Cf. A003961, A156552, A329609, A348717, A353269, A353304 [= a(n^2)].

Programs

PARI

A156552(n) = { my(f = factor(n), p, p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res };
A353269(n) = (!(A156552(n)%3));
A353303(n, m=n) = if(1==n, 1, my(s=0); fordiv(n, d, if((d>1)&&(d<=m)&&A353269(d), s += A353303(n/d, d))); (s));

Formula

a(n) = 0 iff A353269(n) = 0.

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