A024794 -id:A024794 - cp's OEIS frontend

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.

1, 2, 4, 7, 10, 15, 20, 27, 35, 44, 55, 67, 81, 97, 115, 135, 158, 183, 212, 244, 280, 320, 364, 413, 467, 526, 591, 661, 737, 820, 909, 1007, 1112, 1226, 1349, 1481, 1624, 1778, 1943, 2121, 2311, 2515, 2734, 2968, 3219, 3486, 3771, 4075, 4399, 4744, 5112, 5502

Offset: 1

Joseph Foley, Jul 12 2010

nonn

			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.
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.
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.

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 175 terms from Robert G. Wilson v)

Cf. A000586.
Cf. A000070, A024786, A024787, A024788, A024789, A024790, A024791, A024792, A024793, A024794.
Partial sums of A280271.

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
a:= n-> b(n, numtheory[pi](n)): seq(a(n), n=1..80); # Alois P. Heinz

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 *)
(* second program: *)
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]]]];
a[n_] := Sum[k*b[n - k, PrimePi[n - k]], {k, 1, n}];
Table[a[n], {n, 1, 80}] (* Jean-François Alcover, Aug 29 2016, after Alois P. Heinz *)

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

a(n) = Sum_{k=1..n} k * A000586(n-k). - Max Alekseyev, Jul 14 2010

Corrected and extended by R. J. Mathar, Jul 14 2010

A141450 Upper right triangle of the number of m's in all partitions of n.

Original entry on oeis.org

1, 1, 2, 1, 1, 4, 1, 1, 3, 7, 1, 1, 2, 4, 12, 1, 1, 2, 4, 8, 19, 1, 1, 2, 3, 6, 11, 30, 1, 1, 2, 3, 6, 9, 19, 45, 1, 1, 2, 3, 5, 8, 15, 26, 67, 1, 1, 2, 3, 5, 8, 13, 21, 41, 97, 1, 1, 2, 3, 5, 7, 12, 18, 31, 56, 139, 1, 1, 2, 3, 5, 7, 12, 17, 28, 45, 83, 195, 1, 1, 2, 3, 5, 7, 11, 16, 25, 38, 63

Offset: 1

Robert G. Wilson v, Aug 07 2008

The "last" column read from the bottom is A000041.

Mirror of triangle A066633. - Omar E. Pol, May 01 2012

			A000070: 1, 2, 4, 7, 12, 19, 30, 45, 67, 97, 139, 195, 272, 373, 508, ...,
A024786: 0, 1, 1, 3, 4, 8, 11, 19, 26, 41, 56, 83, 112, 160, 213, ...,
A024787: 0, 0, 1, 1, 2, 4, 6, 9, 15, 21, 31, 45, 63, 87, 122, ...,
A024788: 0, 0, 0, 1, 1, 2, 3, 6, 8, 13, 18, 28, 38, 55, 74, ...,
A024789: 0, 0, 0, 0, 1, 1, 2, 3, 5, 8, 12, 17, 25, 35, 50, ...,
A024790: 0, 0, 0, 0, 0, 1, 1, 2, 3, 5, 7, 12, 16, 24, 33, ...,
A024791: 0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 5, 7, 11, 16, 23, ...,
A024792: 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 5, 7, 11, 15, ...,
A024793: 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 5, 7, 11, ...,
A024794: 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 5, 7, ...

Cf. A000041, A000070, A024786, A024787, A024788, A024789, A024790, A024791, A024792, A024793, A024794.

(* First do ) Needs["Combinatorica`"] (* then *) f[n_, m_] := Count[Flatten@ Partitions@ n, m]; Table[ f[n, m], {n, 13}, {m, n, 1, -1}]

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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.

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