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A024938 Total number of parts in all partitions of n into distinct prime parts. 8
0, 1, 1, 0, 3, 0, 3, 2, 2, 5, 1, 5, 3, 5, 5, 7, 5, 10, 6, 10, 12, 10, 15, 12, 16, 17, 17, 19, 22, 17, 27, 21, 30, 30, 31, 35, 36, 40, 45, 45, 49, 53, 50, 62, 60, 69, 69, 73, 78, 85, 88, 98, 100, 105, 116, 116, 134, 135, 141, 149, 154, 168, 176, 188, 195, 206, 211, 232, 242, 255, 267, 276 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,5

LINKS

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

FORMULA

G.f.: sum(x^p(j)/(1+x^p(j)),j>=1)*product(1+x^p(j), j>=1), where p(j) is the j-th prime. - Vladeta Jovovic, Jul 17 2003

EXAMPLE

a(16) = 7 because the partitions of 16 into distinct prime parts are [13,3], [11,5] and [11,3,2].

MAPLE

g:=sum(x^ithprime(j)/(1+x^ithprime(j)), j=1..30)*product(1+x^ithprime(j), j=1..30): gser:=series(g, x=0, 80): seq(coeff(gser, x, n), n=1..75); # Emeric Deutsch, Apr 01 2006

# second Maple program:

with(numtheory):

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

      if n=0 then [1, 0]

    elif i<1 then [0, 0]

    else g:= `if`(ithprime(i)>n, [0$2], b(n-ithprime(i), i-1));

         b(n, i-1) +g +[0, g[1]]

      fi

    end:

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

seq(a(n), n=1..80);  # Alois P. Heinz, Oct 30 2012

MATHEMATICA

Rest@ CoefficientList[ Series[ Sum[x^Prime@j/(1 + x^Prime@j), {j, 20}]* Product[1 + x^Prime@j, {j, 20}], {x, 0, 70}], x] (* Robert G. Wilson v *)

b[n_, i_] := b[n, i] = Module[{g}, If[n==0, {1, 0}, If[i < 1, {0, 0}, g = If[ Prime[i] > n, {0, 0}, b[n - Prime[i], i-1]]; b[n, i-1] + g + {0, g[[1]]}]]]; a[n_] := b[n, PrimePi[n]][[2]]; Table[a[n], {n, 1, 80}] (* Jean-Fran├žois Alcover, Dec 27 2015, after Alois P. Heinz *)

PROG

(PARI)

sumparts(n, pred)={sum(k=1, n, 1 - 1/(1+pred(k)*x^k) + O(x*x^n))*prod(k=1, n, 1+pred(k)*x^k + O(x*x^n))}

{my(n=60); Vec(sumparts(n, isprime), -n)} \\ Andrew Howroyd, Dec 28 2017

CROSSREFS

Cf. A084993.

Sequence in context: A229964 A309722 A070298 * A332715 A219107 A338498

Adjacent sequences:  A024935 A024936 A024937 * A024939 A024940 A024941

KEYWORD

easy,nonn

AUTHOR

Clark Kimberling

EXTENSIONS

More terms from Vladeta Jovovic, Jul 17 2003

STATUS

approved

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Last modified April 22 16:32 EDT 2021. Contains 343177 sequences. (Running on oeis4.)