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

1,16

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, ...

Links

Alois P. Heinz, Table of n, a(n) for n = 1..3000

Wikipedia, Goldbach's weak conjecture

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

Example

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.

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 *)

Crossrefs

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

Sequence in context: A233865 A090552 A024520 * A341446 A015613 A135013

Adjacent sequences: A339998 A339999 A340000 * A340002 A340003 A340004

Keyword

nonn

Author

Michel Lagneau, Dec 26 2020

Status

approved