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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. 1
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

E.g. (i) 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 twenty therefore n7 = 20. E.G. (ii) 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 sixty-seven therefore n12 = 67.

LINKS

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

FORMULA

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

EXAMPLE

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

MAPLE

From R. J. Mathar, Jul 14 2010: (Start)

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:

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

MATHEMATICA

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]

(* 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-Fran├žois Alcover, Aug 29 2016, after Alois P. Heinz *)

PROG

(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

CROSSREFS

Cf. A000070, A024786, A024787, A024788, A024789, A024790, A024791, A024792, A024793, A024794. - Robert G. Wilson v, Jul 14 2010

Sequence in context: A036702 A007983 A049640 * A024668 A188951 A226136

Adjacent sequences:  A179382 A179383 A179384 * A179386 A179387 A179388

KEYWORD

nonn

AUTHOR

Joseph Foley, Jul 12 2010

EXTENSIONS

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

I changed the Mathematica coding to be more efficient Robert G. Wilson v, Jul 20 2010

STATUS

approved

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Last modified March 28 07:54 EDT 2017. Contains 284182 sequences.