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%I #35 Jun 25 2022 02:20:30
%S 1,2,4,7,10,15,20,27,35,44,55,67,81,97,115,135,158,183,212,244,280,
%T 320,364,413,467,526,591,661,737,820,909,1007,1112,1226,1349,1481,
%U 1624,1778,1943,2121,2311,2515,2734,2968,3219,3486,3771,4075,4399,4744,5112,5502
%N The n-th term is the sum of all the 1's generated from all the combinations of prime numbers and ones possible, that add to n, when each prime is only allowed once and any number of ones are allowed.
%H Alois P. Heinz, <a href="/A179385/b179385.txt">Table of n, a(n) for n = 1..10000</a> (first 175 terms from Robert G. Wilson v)
%F a(n) = Sum_{k=1..n} k * A000586(n-k). - _Max Alekseyev_, Jul 14 2010
%e n=7 gives 11111 11, 2111 11, 311 11, 5 11, 5 2, 32 11. (Grouped in 5's) no. of 1's: 7, 5, 4, 2, 0, 2. Sum is 20, therefore a(7) = 20.
%e n=12 gives 11111 11111 11, 11111 11111 2, 11111 311 11, 11111 32 11, 11111 5 11, 5 2111 11, 5 311 11, 5 32 11, 7111 11, 721 11, 73 11, 73 2, 75, eleven 1, no. of 1's: 12, 10, 9, 7, 7, 5, 4, 2, 5, 3, 2, 0, 0, 1. Sum is 67, therefore a(12) = 67.
%e 1: 1 => 1 2: 11, 2 => 2 3: 111, 21 => 4 4: 1111, 211, 22, 31 => 7 5: 11111, 2111, 311, 23 => 10 6: 11111 1, 2111 1, 311 1, 23 1, 5 1 => 15 and so on.
%p b:= proc(n,i) option remember; if n<=0 then 0 elif i=0 then n else b(n, i-1) +b(n-ithprime(i), i-1) fi end: # _R. J. Mathar_, Jul 14 2010
%p a:= n-> b(n, numtheory[pi](n)): seq(a(n), n=1..80); # _Alois P. Heinz_
%t fQ[lst_List] := Sort@ Flatten@ Most@ Split@ lst == Rest@ Union@ lst; f[n_] := Sum[ Count[ Select[ IntegerPartitions[n, {k}, Join[{1}, Prime@ Range@ PrimePi@n]], fQ@# &], 1, 2], {k, n}]; Array[f, 50] (* improved by _Robert G. Wilson v_, Jul 20 2010 *)
%t (* second program: *)
%t b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, b[n, i - 1] + If[Prime[i] > n, 0, b[n - Prime[i], i - 1]]]];
%t a[n_] := Sum[k*b[n - k, PrimePi[n - k]], {k, 1, n}];
%t Table[a[n], {n, 1, 80}] (* _Jean-François Alcover_, Aug 29 2016, after _Alois P. Heinz_ *)
%o (PARI) a(n) = my(r); r = x/(1-x)^2 + O(x^(n+1)); forprime(p=2,n,r*=1+x^p); polcoeff(r,n) \\ _Max Alekseyev_, Jul 14 2010
%Y Cf. A000586.
%Y Cf. A000070, A024786, A024787, A024788, A024789, A024790, A024791, A024792, A024793, A024794.
%Y Partial sums of A280271.
%K nonn
%O 1,2
%A _Joseph Foley_, Jul 12 2010
%E Corrected and extended by _R. J. Mathar_, Jul 14 2010
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