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

1,16

Comments

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

Links

Antti Karttunen, Table of n, a(n) for n = 1..65537

Index entries for sequences computed from indices in prime factorization

Formula

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

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

Example

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.

Prog

(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));

Crossrefs

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

Sequence in context: A333817 A270417 A353333 * A307666 A319995 A266344

Adjacent sequences: A353300 A353301 A353302 * A353304 A353305 A353306

Keyword

nonn

Author

Antti Karttunen, Apr 10 2022

Status

approved