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%I A333615 #6 May 16 2020 03:46:41
%S A333615 1,2,3,4,7,10,13,20,26,34,48,61,78,103,129,162,206,256,314,391,479,
%T A333615 579,711,859,1028,1243,1485,1764,2107,2497,2941,3477,4092,4783,5610,
%U A333615 6557,7615,8872,10303,11901,13781,15910,18292,21062,24196,27697,31726,36287
%N 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.
%e A333615 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):
%e A333615 7 = 5 + 1 + 1
%e A333615 7 = 3 + 3 + 1
%e A333615 7 = 3 + 1 + 1 + 1 + 1
%e A333615 7 = 1 + 1 + 1 + 1 + 1 + 1 + 1
%e A333615 So, a(3) = 4.
%t A333615 a[n_] := Module[{p},
%t A333615   p = Table[Prime[i] - 2, {i, 2, PrimePi[2*n + 3]}];
%t A333615   Length[IntegerPartitions[2*n + 1, {0, Infinity}, p]]]
%t A333615 Table[a[n], {n, 0, 60}]
%o A333615 (PARI)
%o A333615 \\ Slowish:
%o A333615 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)));
%o A333615 odd_primes_minus2_upto_n_reversed(n) = { my(lista=List([])); forprime(p=3,n+2,listput(lista,p-2)); Vecrev(Vec(lista)); };
%o A333615 A333615(n) = partitions_into(n+n+1, odd_primes_minus2_upto_n_reversed(n+n+1)); \\ _Antti Karttunen_, May 09 2020
%Y A333615 Cf. A069259 (partitions of 2*n, instead of 2*n+1).
%Y A333615 Cf. A101776.
%K A333615 nonn
%O A333615 0,2
%A A333615 _Luc Rousseau_, Mar 29 2020

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