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

1,36

Links

T. D. Noe, Table of n, a(n) for n = 1..10000

Eric Weisstein's World of Mathematics, Prime Sums

Formula

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

Example

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

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 *)

Crossrefs

Cf. A084146, A084147.

Sequence in context: A349433 A346478 A346250 * A025888 A145708 A138532

Adjacent sequences: A084140 A084141 A084142 * A084144 A084145 A084146

Keyword

nonn

Author

Eric W. Weisstein, May 15 2003

Status

approved