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A323016 a(n) is the number of ordered partitions of 24*n + 4 into four squares of primes (A001248). 1
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 (list; graph; refs; listen; history; text; internal format)
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 A091651 A199865

Adjacent sequences:  A323013 A323014 A323015 * A323017 A323018 A323019

KEYWORD

nonn

AUTHOR

Jianing Song, Jan 05 2019

STATUS

approved

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Last modified January 19 18:25 EST 2020. Contains 331051 sequences. (Running on oeis4.)