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

0,2

Links

Table of n, a(n) for n=0..47.

Example

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.

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}]

Prog

(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

Crossrefs

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

Cf. A101776.

Sequence in context: A065461 A008824 A261616 * A329774 A081942 A228588

Adjacent sequences: A333612 A333613 A333614 * A333616 A333617 A333618

Keyword

nonn

Author

Luc Rousseau, Mar 29 2020

Status

approved