A036497 Number of partitions of n into distinct primes (counting 1 as a prime).

1, 1, 1, 2, 1, 2, 2, 2, 3, 2, 3, 3, 3, 4, 4, 4, 5, 5, 6, 7, 7, 8, 8, 9, 10, 10, 11, 11, 11, 13, 13, 15, 16, 16, 18, 18, 20, 22, 22, 24, 25, 26, 29, 30, 32, 33, 34, 37, 39, 41, 44, 45, 47, 51, 53, 57, 59, 61, 64, 67, 72, 76, 79, 82, 86, 89, 95, 100, 103, 108, 112, 118

Offset: 0

Author

Wouter Meeussen, Dec 17 1998

Keywords

• nonn

Comments

Honsberger shows that the primes-including-1 are a complete sequence and therefore all numbers in this sequence exceed zero. - Ron Knott, Aug 27 2016

Number of partitions of n into distinct noncomposite numbers. - Omar E. Pol, Dec 14 2024

Examples

			a(11) = 3 since 11 = 1+2+3+5=1+3+7 has 3 partitions of distinct primes-including-1. - _Ron Knott_, Aug 27 2016
		

References

• Ross Honsberger, Mathematical Gems III, The Mathematical Association of America, 1985, pages 127-128.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A000586, A008578.

Programs

Maple

s:= proc(n) option remember;
      `if`(n<1, n+1, ithprime(n)+s(n-1))
    end:
b:= proc(n, i) option remember; (p-> `if`(n=0, 1,
      `if`(n>s(i), 0, b(n, i-1)+ `if`(p>n, 0,
       b(n-p, i-1)))))(`if`(i<1, 1, ithprime(i)))
    end:
a:= n-> b(n, numtheory[pi](n)):
seq(a(n), n=0..100);  # Alois P. Heinz, Aug 27 2016

Mathematica

myprime[ n_ ] := If[ n===0, 1, Prime[ n ] ]; ta1=Table[ Product[ 1+z^myprime[ k ], {k, 0, n} ]~CoefficientList~z, {n, 31, 32} ]; leveled=Count[ Take[ Last@ta1, Length@ta1[ [ -2 ] ] ]-ta1[ [ -2 ] ], 0 ]; Take[ Last@ta1, leveled ]
Table[Length@ DeleteCases[DeleteCases[IntegerPartitions@ n, {_, a_, _} /; CompositeQ@ a], w_ /; MemberQ[Differences@ w, 0]], {n, 0, 60}] (* Michael De Vlieger, Aug 27 2016 *)

Formula

G.f.: (1 + x)*Product_{k>=1} (1 + x^prime(k)). - Ilya Gutkovskiy, Dec 31 2016