A323016 a(n) is the number of ordered partitions of 24*n + 4 into four squares of primes (A001248).

0, 0, 0, 0, 1, 4, 6, 4, 5, 12, 16, 16, 18, 16, 18, 28, 34, 28, 26, 36, 49, 40, 44, 52, 42, 52, 70, 52, 47, 60, 76, 72, 54, 76, 60, 48, 88, 68, 50, 72, 78, 80, 48, 96, 102, 60, 98, 76, 79, 96, 104, 112, 52, 108, 132, 64, 112, 88, 94, 120, 89, 136, 72, 88, 168, 96

Offset

0,6

Comments

The main entry is A323015, which is the unordered version.

Also, a(n) is the number of ordered partitions of n into four terms of A024702.

a(n) > 0 for 4 <= n <= 2*10^4. Conjecture: a(n) > 0 for all n >= 4. A stronger conjecture: lim inf a(n) = +oo.

Links

Table of n, a(n) for n=0..65.

Formula

G.f.: (Sum_{primes p>=5} x^((p^2-1)/24))^4 = (Sum_{k>=3} x^A024702(k))^4.

Example

100 = 5^2 + 5^2 + 5^2 + 5^2 (1 permutation).
124 = 5^2 + 5^2 + 5^2 + 7^2 (4 permutations).
148 = 5^2 + 5^2 + 7^2 + 7^2 (6 permutations).
172 = 5^2 + 7^2 + 7^2 + 7^2 (4 permutations).
196 = 7^2 + 7^2 + 7^2 + 7^2 (1 permutation) = 5^2 + 5^2 + 5^2 + 11^2 (4 permutations).
220 = 5^2 + 5^2 + 7^2 + 11^2 (12 permutations).
244 = 5^2 + 7^2 + 7^2 + 11^2 (12 permutations) = 5^2 + 5^2 + 5^2 + 13^2 (4 permutations).
268 = 7^2 + 7^2 + 7^2 + 11^2 (4 permutations) = 5^2 + 5^2 + 7^2 + 13^2 (12 permutations).
...

Prog

(PARI) a(n) = if(n<4, 0, polcoeff(sum(p=5, sqrt(24*n-48), if(isprime(p), x^((p^2-1)/24), 0))^4, n))

Crossrefs

Cf. A001248, A024702, A323015.

Sequence in context: A123999 A014110 A266491 * A261637 A366558 A091651

Adjacent sequences: A323013 A323014 A323015 * A323017 A323018 A323019

Keyword

nonn

Author

Jianing Song, Jan 05 2019

Status

approved