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

Author

Amarnath Murthy, Mar 19 2002

Keywords

• nonn

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

Examples

			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.
		

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

• David A. Corneth, Table of n, a(n) for n = 1..10000 (first 200 terms from Jean-François Alcover)
• Jean-François Alcover, Conjectured terms up to a(200).

Crossrefs

Cf. A102330, A013918, A007504.

Programs

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

PARI

a(n)=
{
    my(P=primes(n), k=n, t, res = oo);
    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),
		res = min(res, t);
	   )
        ,
            2 \\ flag: only strictly increasing vectors v
        );
        P=concat(P, nextprime(P[k]+1));
        k++;
	if(P[k] + sum(i = 1+bitand(n,1), n-1+bitand(n,1), P[i]) > res,
		return(res)
	)
    );
}
\\ Charles R Greathouse IV, Sep 19 2015; corrected by David A. Corneth, May 12 2025

Formula

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

Extensions

More terms from Sascha Kurz, Feb 03 2003

Corrected by Ray Chandler, Feb 02 2005