A068873 Smallest prime which is a sum of n distinct primes.

2, 5, 19, 17, 43, 41, 79, 83, 127, 131, 199, 197, 283, 281, 379, 389, 499, 509, 643, 641, 809, 809, 983, 971, 1171, 1163, 1381, 1373, 1609, 1607, 1861, 1861, 2137, 2137, 2437, 2441, 2749, 2767, 3109, 3109, 3457, 3457, 3833, 3847, 4243, 4241, 4663, 4679, 5119

Offset

1,1

Comments

Conjectured terms a(50)-a(76): 5147, 5623, 5591, 6079, 6101, 6599, 6607, 7151, 7151, 7699, 7699, 8273, 8293, 8893, 8893, 9521, 9547, 10211, 10223, 10889, 10891, 11597, 11617, 12343, 12373, 13099, 13127. - Jean-François Alcover, Apr 22 2020

References

Shantanu Dey & Moloy De, Two conjectures on prime numbers, Journal of Recreational Mathematics, Vol. 36 (3), pp 205-206. Baywood Publ. Co, Amityville NY 2011.

Links

Table of n, a(n) for n=1..49.

Jean-François Alcover, Conjectured terms up to a(200).

Formula

Min(a(n), A073619(n)) = A007504(n) for n > 1. - Jonathan Sondow, Jul 10 2012

Example

a(3) = 19 as 19 is the smallest prime which can be expressed as the sum of three primes as 19 = 3 + 5 + 11. a(4) = 17= 2+3+5+7. a(2)=A038609(1). a(3)=A124867(7). Further examples in A102330.

Maple

# Number of ways to write n as a sum of k distinct primes, the smallest
# being smalp
sumkprims := proc(n, k, smalp)
    option remember;
    local a, res, pn;
    res := n-smalp ;
    if res < 0 then
        return 0;
    elif res > 0 and k <=0 then
        return 0;
    elif res = 0 and k = 1 then
        return 1;
    else
        pn := nextprime(smalp) ;
        a := 0 ;
        while pn <= res do
            a := a+procname(res, k-1, pn) ;
            pn := nextprime(pn) ;
        end do:
        a ;
    end if;
end proc:
# Number of ways of writing n as a sum of k distinct primes
A000586k := proc(n, k)
    local a, i, smalp ;
    a := 0 ;
    for i from 1 do
        smalp := ithprime(i) ;
        if k*smalp > n then
            return a;
        end if;
        a := a+sumkprims(n, k, smalp) ;
    end do:
end proc:
# Smallest prime which is a sum of n distinct primes
A068873 := proc(n)
    local a, i;
    a := A007504(n) ;
    a := nextprime(a-1) ;
    for i from 1 do
        if A000586k(a, n) > 0 then
            return a;
        end if;
        a := nextprime(a) ;
    end do:
end proc: # R. J. Mathar, May 04 2014

Prog

(PARI) a(n)=
{
    my(P=primes(n), k=n, t);
    while(1,
        forvec(v=vector(n-1, i, [1, k-1]),
            t=sum(i=1, n-1, P[v[i]])+P[k];
            if(isprime(t), return(t))
        ,
            2 \\ flag: only strictly increasing vectors v
        );
        P=concat(P, nextprime(P[k]+1));
        k++
    );
} \\ Charles R Greathouse IV, Sep 19 2015

Crossrefs

Cf. A102330, A013918, A007504.

Sequence in context: A356366 A323706 A125765 * A035091 A045367 A045368

Adjacent sequences: A068870 A068871 A068872 * A068874 A068875 A068876

Keyword

nonn

Author

Amarnath Murthy, Mar 19 2002

Extensions

More terms from Sascha Kurz, Feb 03 2003

Corrected by Ray Chandler, Feb 02 2005

Status

approved