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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
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.
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
KEYWORD
nonn
AUTHOR
Jianing Song, Jan 05 2019
STATUS
approved