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A144119 Total number of nonprime parts in all partitions of n. 5
1, 2, 4, 8, 13, 22, 34, 54, 80, 119, 170, 246, 342, 478, 653, 894, 1198, 1610, 2127, 2813, 3672, 4789, 6181, 7975, 10192, 13010, 16488, 20861, 26224, 32918, 41086, 51199, 63494, 78599, 96888, 119235, 146167, 178879, 218181, 265662, 322487, 390834, 472343 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

a(n) is also the sum of the differences between the sum of m-th largest and the sum of (m+1)st largest elements in all partitions of n for all nonprimes m. - Omar E. Pol, Oct 27 2012

LINKS

Alois P. Heinz, Table of n, a(n) for n = 1..1000

FORMULA

a(n) = A006128(n)-A037032(n).

EXAMPLE

From Omar E. Pol, Nov 20 2011 (Start):

For n = 6 we have:

--------------------------------------

.                        Number of

Partitions            nonprime parts

--------------------------------------

6 .......................... 1

3 + 3 ...................... 0

4 + 2 ...................... 1

2 + 2 + 2 .................. 0

5 + 1 ...................... 1

3 + 2 + 1 .................. 1

4 + 1 + 1 .................. 3

2 + 2 + 1 + 1 .............. 2

3 + 1 + 1 + 1 .............. 3

2 + 1 + 1 + 1 + 1 .......... 4

1 + 1 + 1 + 1 + 1 + 1 ...... 6

------------------------------------

Total ..................... 22

So a(6) = 22. (End)

MAPLE

b:= proc(n, i) option remember; local g;

      if n=0 then [1, 0]

    elif i<1 then [0, 0]

    else g:= `if`(i>n, [0$2], b(n-i, i));

         b(n, i-1) +g +[0, `if`(isprime(i), 0, g[1])]

      fi

    end:

a:= n-> b(n, n)[2]:

seq(a(n), n=1..100);  # Alois P. Heinz, Oct 27 2012

MATHEMATICA

b[n_, i_] := b[n, i] = Module[{g}, If[n == 0, {1, 0}, If[i<1, {0, 0}, g = If[i>n, {0, 0}, b[n-i, i]]; b[n, i-1] + g + {0, If[PrimeQ[i], 0, g[[1]]]} ]]]; a[n_] := b[n, n][[2]]; Table[a[n], {n, 1, 100}] (* Jean-Fran├žois Alcover, Oct 29 2015, after Alois P. Heinz *)

PROG

(PARI) vector(100, n, sum(k=1, n, (numdiv(k)-omega(k))*numbpart(n-k))) \\ Altug Alkan, Oct 29 2015

CROSSREFS

Cf. A006128, A018252, A037032, A144116, A144121.

Sequence in context: A164480 A134035 A078157 * A207033 A291553 A330153

Adjacent sequences:  A144116 A144117 A144118 * A144120 A144121 A144122

KEYWORD

easy,nonn

AUTHOR

Omar E. Pol, Sep 11 2008

STATUS

approved

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Last modified April 18 07:59 EDT 2021. Contains 343084 sequences. (Running on oeis4.)