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A376348 a(n) is the number of multisets with n primes with which an n-gon with perimeter prime(n) can be formed.

%I A376348 #7 Oct 25 2024 09:49:32
%S A376348 0,0,1,1,2,2,3,7,7,12,19,19,25,44,72,72,119,147,152,234,292,435,777,
%T A376348 920,946,1135,1161,1377,3702,4293,5942,5942,10741,10741,14483,18953,
%U A376348 22091,28658,37686,37686,63053,63053,72389,72389,132732,233773,265312,265312,300443,373266
%N A376348 a(n) is the number of multisets with n primes with which an n-gon with perimeter prime(n) can be formed.
%C A376348 a(n) is the number of partitions of prime(n) into n prime parts < prime(n)/2.
%C A376348 First differs from A259254 at n=31: a(31) = 3702 but A259254(31) = 3703.
%H A376348 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Polygon.html">Polygon</a>
%e A376348 a(7) = 2 because exactly the 2 partitions (2, 2, 2, 2, 3, 3, 3) and (2, 2, 2, 2, 2, 2, 5) have 7 prime parts and their sum is p(7) = 17.
%p A376348 A376348:=proc(n)
%p A376348    local a,p,x,i;
%p A376348    a:=0;
%p A376348    p:=ithprime(n);
%p A376348    for x from NumberTheory:-pi(p/n)+1 to NumberTheory:-pi(p/2) do
%p A376348       a:=a+numelems(select(i->nops(i)=n-1 and andmap(isprime,i),combinat:-partition(ithprime(n)-ithprime(x),ithprime(x))))
%p A376348    od;
%p A376348    return a
%p A376348 end proc;
%p A376348 seq(A376348(n),n=3..42);
%o A376348 (PARI) a(n)={my(m=prime(n), p=primes(primepi((m-1)\2))); polcoef(polcoef(1/prod(i=1, #p, 1 - y*x^p[i], 1 + O(x*x^m)), m),n)} \\ _Andrew Howroyd_, Oct 13 2024
%Y A376348 Cf. A005044, A007042, A038499, A062890, A069906, A069907, A259254, A288253, A288254, A288255, A288256, A318604, A366398.
%K A376348 nonn
%O A376348 3,5
%A A376348 _Felix Huber_, Oct 13 2024
%E A376348 a(43) onwards from _Andrew Howroyd_, Oct 13 2024