A218007 Number of partitions of n into at most three primes (including 1).

1, 2, 3, 3, 4, 4, 5, 4, 5, 4, 5, 4, 6, 5, 7, 5, 7, 5, 8, 6, 9, 6, 9, 7, 10, 7, 10, 5, 10, 6, 12, 7, 13, 7, 12, 8, 14, 7, 14, 6, 15, 8, 17, 9, 17, 8, 18, 10, 19, 10, 19, 7, 20, 9, 21, 9, 20, 7, 21, 11, 25, 11, 24, 9, 26, 11, 27, 9, 24, 8, 28, 12, 30, 13, 29

Offset: 1

Author

Frank M Jackson, Mar 26 2013

Keywords

• nonn

Comments

The above sequence relies on the strong Goldbach's conjecture that any positive integer is the sum of at most three distinct terms from {1 union primes}.

Examples

			a(21)=9 as 21 = 1+1+19 = 2+19 = 1+3+17 = 2+2+17 = 1+7+13 = 3+5+13 = 3+7+11 = 5+5+11 = 7+7+7
		

Links

• T. D. Noe, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A002375, A025583, A068307, A071335, A185101.

Programs

Mathematica

primeQ[p0_] := If[p0==1, True, PrimeQ[p0]]; SetAttributes[primeQ, Listable]; goldbachcount[p1_] := (parts=IntegerPartitions[p1, 3]; count=0; n=1; While[n<=Length[parts], If[Intersection[Flatten[primeQ[parts[[n]]]]][[1]] == True, count++]; n++]; count); Table[goldbachcount[i], {i, 1, 100}]
Table[Length[Select[#/.(1->2)&/@IntegerPartitions[n,3],AllTrue[#,PrimeQ]&]],{n,80}] (* Harvey P. Dale, Jan 11 2023 *)