cp's OEIS Frontend

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

%I A340001 #32 Jul 07 2024 17:13:20
%S A340001 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,
%T A340001 109,169,198,235,240,322,331,379,437,497,565,635,634,803,798,896,888,
%U A340001 1091,1328,1477,1444,1616,1753,1730,2080,2262,2452,2627,2588,2790,3043,3004,3535
%N A340001 Number of ways prime(n) is a sum of five distinct primes.
%C A340001 Conjecture: all primes >= 43 are the sum of five distinct primes.
%C A340001 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, ...
%C A340001 The primes in the sequence are 2, 5, 11, 29, 109, 331, 379, 1091, 1753, ...
%C A340001 The squares in the sequence are 0, 1, 16, 25, 169, 1444, ...
%H A340001 Alois P. Heinz, <a href="/A340001/b340001.txt">Table of n, a(n) for n = 1..3000</a>
%H A340001 Wikipedia, <a href="http://en.wikipedia.org/wiki/Goldbach&#39;s_weak_conjecture">Goldbach's weak conjecture</a>
%F A340001 a(n) = A219199(A000040(n)).
%F A340001 a(n) = [x^prime(n)*y^5] Product_{i>=1} (1+x^prime(i)*y). - _Alois P. Heinz_, Dec 30 2020
%e A340001 a(14) = 1 because prime(14) = 43 = 3 + 5 + 7 + 11 + 17.
%e A340001 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.
%p A340001 b:= proc(n, i) option remember; series(`if`(n=0, 1,
%p A340001       `if`(i<1, 0, b(n, i-1)+(p-> `if`(p>n, 0,
%p A340001        x*b(n-p, i-1)))(ithprime(i)))), x, 6)
%p A340001     end:
%p A340001 a:= n-> coeff(b(ithprime(n), n), x, 5):
%p A340001 seq(a(n), n=1..100);  # _Alois P. Heinz_, Dec 30 2020
%t A340001 b[n_, i_] := b[n, i] = Series[If[n == 0, 1,
%t A340001      If[i < 1, 0, b[n, i - 1] + Function[p, If[p > n, 0,
%t A340001      x*b[n - p, i - 1]]][Prime[i]]]], {x, 0, 6}];
%t A340001 a[n_] := SeriesCoefficient[b[Prime[n], n], {x, 0, 5}];
%t A340001 Array[a, 100] (* _Jean-François Alcover_, Apr 26 2021, after _Alois P. Heinz_ *)
%t A340001 Table[Length[Select[IntegerPartitions[p,{5}],AllTrue[#,PrimeQ]&&Length[Union[#]]==5&]],{p,Prime[Range[70]]}] (* _Harvey P. Dale_, Jul 07 2024 *)
%Y A340001 Cf. A000040, A002372, A002375, A024684, A062301, A178041, A219199, A224534.
%K A340001 nonn
%O A340001 1,16
%A A340001 _Michel Lagneau_, Dec 26 2020