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A179385 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.

%I A179385 #35 Jun 25 2022 02:20:30
%S A179385 1,2,4,7,10,15,20,27,35,44,55,67,81,97,115,135,158,183,212,244,280,
%T A179385 320,364,413,467,526,591,661,737,820,909,1007,1112,1226,1349,1481,
%U A179385 1624,1778,1943,2121,2311,2515,2734,2968,3219,3486,3771,4075,4399,4744,5112,5502
%N A179385 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 A179385 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 A179385 a(n) = Sum_{k=1..n} k * A000586(n-k). - _Max Alekseyev_, Jul 14 2010
%e A179385 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 A179385 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 A179385 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 A179385 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 A179385 a:= n-> b(n, numtheory[pi](n)): seq(a(n), n=1..80); # _Alois P. Heinz_
%t A179385 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 A179385 (* second program: *)
%t A179385 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 A179385 a[n_] := Sum[k*b[n - k, PrimePi[n - k]], {k, 1, n}];
%t A179385 Table[a[n], {n, 1, 80}] (* _Jean-François Alcover_, Aug 29 2016, after _Alois P. Heinz_ *)
%o A179385 (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 A179385 Cf. A000586.
%Y A179385 Cf. A000070, A024786, A024787, A024788, A024789, A024790, A024791, A024792, A024793, A024794.
%Y A179385 Partial sums of A280271.
%K A179385 nonn
%O A179385 1,2
%A A179385 _Joseph Foley_, Jul 12 2010
%E A179385 Corrected and extended by _R. J. Mathar_, Jul 14 2010