A084143 Number of partitions of n into a sum of two or more consecutive primes.

0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 0, 0, 0, 0, 2, 0, 0, 1, 0, 2, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 2, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 2, 0, 0, 1

Offset: 1

Author

Eric W. Weisstein, May 15 2003

Keywords

• nonn

Examples

			a(36)=2 because we have 36 = 17 + 19 = 5 + 7 + 11 + 13.
		

Links

• T. D. Noe, Table of n, a(n) for n = 1..10000
• Eric Weisstein's World of Mathematics, Prime Sums

Crossrefs

Cf. A084146, A084147.

Programs

Maple

g:=sum(sum(product(x^ithprime(k),k=i..j),j=i+1..25),i=1..25): gser:=series(g,x=0,80): seq(coeff(gser,x,n),n=1..75); # Emeric Deutsch, Mar 30 2006
# alternative, R. J. Mathar, Aug 19 2020
A084143 := proc(n::integer)
    local a,k,i,spr ;
    a := 0 ;
    for k from 2 do
        if add(ithprime(i),i=1..k) > n then
            break;
        end if;
        for i from 1 do
            spr := add( ithprime(j),j=i..i+k-1) ;
            if spr > n then
                break;
            end if;
            if spr = n then
                a := a +1 ;
            end if;
        end do:
    end do:
    a ;
end proc:

Mathematica

max = 25; gf = Sum[ Sum[ Product[ x^Prime[k], {k, i, j}], {j, i+1, max}], {i, 1, max}]; Rest[ CoefficientList[gf, x]][[1 ;; 75]] (* Jean-François Alcover, Oct 23 2012, after Emeric Deutsch *)

Formula

G.f.: Sum_{i>=1} Sum_{j>=i+1} Product_{k=i..j} x^prime(k). - Emeric Deutsch, Mar 30 2006