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%I #30 Apr 26 2021 07:31:47
%S 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 109,169,198,235,240,322,331,379,437,497,565,635,634,803,798,896,888,
%U 1091,1328,1477,1444,1616,1753,1730,2080,2262,2452,2627,2588,2790,3043,3004,3535
%N Number of ways prime(n) is a sum of five distinct primes.
%C Conjecture: all primes >= 43 are the sum of five distinct primes.
%C 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 The primes in the sequence are 2, 5, 11, 29, 109, 331, 379, 1091, 1753, ...
%C The squares in the sequence are 0, 1, 16, 25, 169, 1444, ...
%H Alois P. Heinz, <a href="/A340001/b340001.txt">Table of n, a(n) for n = 1..3000</a>
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Goldbach's_weak_conjecture">Goldbach's weak conjecture</a>
%F a(n) = A219199(A000040(n)).
%F a(n) = [x^prime(n)*y^5] Product_{i>=1} (1+x^prime(i)*y). - _Alois P. Heinz_, Dec 30 2020
%e a(14) = 1 because prime(14) = 43 = 3 + 5 + 7 + 11 + 17.
%e 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 b:= proc(n, i) option remember; series(`if`(n=0, 1,
%p `if`(i<1, 0, b(n, i-1)+(p-> `if`(p>n, 0,
%p x*b(n-p, i-1)))(ithprime(i)))), x, 6)
%p end:
%p a:= n-> coeff(b(ithprime(n), n), x, 5):
%p seq(a(n), n=1..100); # _Alois P. Heinz_, Dec 30 2020
%t b[n_, i_] := b[n, i] = Series[If[n == 0, 1,
%t If[i < 1, 0, b[n, i - 1] + Function[p, If[p > n, 0,
%t x*b[n - p, i - 1]]][Prime[i]]]], {x, 0, 6}];
%t a[n_] := SeriesCoefficient[b[Prime[n], n], {x, 0, 5}];
%t Array[a, 100] (* _Jean-François Alcover_, Apr 26 2021, after _Alois P. Heinz_ *)
%Y Cf. A000040, A002372, A002375, A024684, A062301, A178041, A219199, A224534.
%K nonn
%O 1,16
%A _Michel Lagneau_, Dec 26 2020
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