proposed

approved

editing

proposed

(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

proposed

editing

editing

proposed

allocated for Luc Rousseaua(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

0,2

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.

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}]

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

Cf. A101776.

allocated

nonn

Luc Rousseau, Mar 29 2020

approved

editing

allocated for Luc Rousseau

allocated

approved