A333615 a(n) is the number of ways to express 2*n+1 as a sum of parts x such that x+2 is an odd prime.

1, 2, 3, 4, 7, 10, 13, 20, 26, 34, 48, 61, 78, 103, 129, 162, 206, 256, 314, 391, 479, 579, 711, 859, 1028, 1243, 1485, 1764, 2107, 2497, 2941, 3477, 4092, 4783, 5610, 6557, 7615, 8872, 10303, 11901, 13781, 15910, 18292, 21062, 24196, 27697, 31726, 36287

Offset: 0

Author

Luc Rousseau, Mar 29 2020

Keywords

• nonn

Examples

			For n = 3, 2*n + 1 = 7. There are 4 partitions of 7 into parts with sizes 1, 3, 5, 9, 11 ... (the odd primes minus 2):
7 = 5 + 1 + 1
7 = 3 + 3 + 1
7 = 3 + 1 + 1 + 1 + 1
7 = 1 + 1 + 1 + 1 + 1 + 1 + 1
So, a(3) = 4.
		

Crossrefs

Cf. A069259 (partitions of 2*n, instead of 2*n+1).

Cf. A101776.

Programs

Mathematica

a[n_] := Module[{p},
  p = Table[Prime[i] - 2, {i, 2, PrimePi[2*n + 3]}];
  Length[IntegerPartitions[2*n + 1, {0, Infinity}, p]]]
Table[a[n], {n, 0, 60}]

PARI

\\ Slowish:
partitions_into(n,parts,from=1) = if(!n,1, if(#parts==from, (0==(n%parts[from])), my(s=0); for(i=from,#parts,if(parts[i]<=n, s += partitions_into(n-parts[i],parts,i))); (s)));
odd_primes_minus2_upto_n_reversed(n) = { my(lista=List([])); forprime(p=3,n+2,listput(lista,p-2)); Vecrev(Vec(lista)); };
A333615(n) = partitions_into(n+n+1, odd_primes_minus2_upto_n_reversed(n+n+1)); \\ Antti Karttunen, May 09 2020