cp's OEIS Frontend

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

%I A323016 #11 Jan 11 2019 17:55:48
%S A323016 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 A323016 70,52,47,60,76,72,54,76,60,48,88,68,50,72,78,80,48,96,102,60,98,76,
%U A323016 79,96,104,112,52,108,132,64,112,88,94,120,89,136,72,88,168,96
%N A323016 a(n) is the number of ordered partitions of 24*n + 4 into four squares of primes (A001248).
%C A323016 The main entry is A323015, which is the unordered version.
%C A323016 Also, a(n) is the number of ordered partitions of n into four terms of A024702.
%C A323016 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 A323016 G.f.: (Sum_{primes p>=5} x^((p^2-1)/24))^4 = (Sum_{k>=3} x^A024702(k))^4.
%e A323016 100 = 5^2 + 5^2 + 5^2 + 5^2 (1 permutation).
%e A323016 124 = 5^2 + 5^2 + 5^2 + 7^2 (4 permutations).
%e A323016 148 = 5^2 + 5^2 + 7^2 + 7^2 (6 permutations).
%e A323016 172 = 5^2 + 7^2 + 7^2 + 7^2 (4 permutations).
%e A323016 196 = 7^2 + 7^2 + 7^2 + 7^2 (1 permutation) = 5^2 + 5^2 + 5^2 + 11^2 (4 permutations).
%e A323016 220 = 5^2 + 5^2 + 7^2 + 11^2 (12 permutations).
%e A323016 244 = 5^2 + 7^2 + 7^2 + 11^2 (12 permutations) = 5^2 + 5^2 + 5^2 + 13^2 (4 permutations).
%e A323016 268 = 7^2 + 7^2 + 7^2 + 11^2 (4 permutations) = 5^2 + 5^2 + 7^2 + 13^2 (12 permutations).
%e A323016 ...
%o A323016 (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 A323016 Cf. A001248, A024702, A323015.
%K A323016 nonn
%O A323016 0,6
%A A323016 _Jianing Song_, Jan 05 2019