A340001 Number of ways prime(n) is a sum of five distinct primes.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 5, 6, 11, 14, 16, 25, 29, 39, 57, 68, 75, 88, 92, 109, 169, 198, 235, 240, 322, 331, 379, 437, 497, 565, 635, 634, 803, 798, 896, 888, 1091, 1328, 1477, 1444, 1616, 1753, 1730, 2080, 2262, 2452, 2627, 2588, 2790, 3043, 3004, 3535

Offset: 1

Author

Michel Lagneau, Dec 26 2020

Keywords

• nonn

Comments

Conjecture: all primes >= 43 are the sum of five distinct primes.

The sequence of the prime numbers that are the sum of five distinct prime numbers begins with 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, ...

The primes in the sequence are 2, 5, 11, 29, 109, 331, 379, 1091, 1753, ...

The squares in the sequence are 0, 1, 16, 25, 169, 1444, ...

Examples

			a(14) = 1 because prime(14) = 43 = 3 + 5 + 7 + 11 + 17.
a(17) = 5 because prime(17) = 59 = 3 + 5 + 7 + 13 + 31 = 3 + 5 + 11 + 17 + 23 = 3 + 7 + 13 + 17 + 19 = 5 + 7 + 11 + 13 + 23 = 5 + 7 + 11 + 17 + 19.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..3000
• Wikipedia, Goldbach's weak conjecture

Crossrefs

Cf. A000040, A002372, A002375, A024684, A062301, A178041, A219199, A224534.

Programs

Maple

b:= proc(n, i) option remember; series(`if`(n=0, 1,
      `if`(i<1, 0, b(n, i-1)+(p-> `if`(p>n, 0,
       x*b(n-p, i-1)))(ithprime(i)))), x, 6)
    end:
a:= n-> coeff(b(ithprime(n), n), x, 5):
seq(a(n), n=1..100);  # Alois P. Heinz, Dec 30 2020

Mathematica

b[n_, i_] := b[n, i] = Series[If[n == 0, 1,
     If[i < 1, 0, b[n, i - 1] + Function[p, If[p > n, 0,
     x*b[n - p, i - 1]]][Prime[i]]]], {x, 0, 6}];
a[n_] := SeriesCoefficient[b[Prime[n], n], {x, 0, 5}];
Array[a, 100] (* Jean-François Alcover, Apr 26 2021, after Alois P. Heinz *)
Table[Length[Select[IntegerPartitions[p,{5}],AllTrue[#,PrimeQ]&&Length[Union[#]]==5&]],{p,Prime[Range[70]]}] (* Harvey P. Dale, Jul 07 2024 *)

Formula

a(n) = A219199(A000040(n)).

a(n) = [x^prime(n)*y^5] Product_{i>=1} (1+x^prime(i)*y). - Alois P. Heinz, Dec 30 2020