A323016 - OEIS

%I #11 Jan 11 2019 17:55:48

%S 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,

%T 70,52,47,60,76,72,54,76,60,48,88,68,50,72,78,80,48,96,102,60,98,76,

%U 79,96,104,112,52,108,132,64,112,88,94,120,89,136,72,88,168,96

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

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

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

%C 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.

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

%e 100 = 5^2 + 5^2 + 5^2 + 5^2 (1 permutation).

%e 124 = 5^2 + 5^2 + 5^2 + 7^2 (4 permutations).

%e 148 = 5^2 + 5^2 + 7^2 + 7^2 (6 permutations).

%e 172 = 5^2 + 7^2 + 7^2 + 7^2 (4 permutations).

%e 196 = 7^2 + 7^2 + 7^2 + 7^2 (1 permutation) = 5^2 + 5^2 + 5^2 + 11^2 (4 permutations).

%e 220 = 5^2 + 5^2 + 7^2 + 11^2 (12 permutations).

%e 244 = 5^2 + 7^2 + 7^2 + 11^2 (12 permutations) = 5^2 + 5^2 + 5^2 + 13^2 (4 permutations).

%e 268 = 7^2 + 7^2 + 7^2 + 11^2 (4 permutations) = 5^2 + 5^2 + 7^2 + 13^2 (12 permutations).

%e ...

%o (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))

%Y Cf. A001248, A024702, A323015.

%K nonn

%O 0,6

%A _Jianing Song_, Jan 05 2019