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Smallest prime which is a sum of n distinct primes.
8

%I #42 Apr 28 2020 08:54:24

%S 2,5,19,17,43,41,79,83,127,131,199,197,283,281,379,389,499,509,643,

%T 641,809,809,983,971,1171,1163,1381,1373,1609,1607,1861,1861,2137,

%U 2137,2437,2441,2749,2767,3109,3109,3457,3457,3833,3847,4243,4241,4663,4679,5119

%N Smallest prime which is a sum of n distinct primes.

%C 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

%D 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.

%H Jean-François Alcover, <a href="/A068873/a068873.txt">Conjectured terms up to a(200).</a>

%F Min(a(n), A073619(n)) = A007504(n) for n > 1. - _Jonathan Sondow_, Jul 10 2012

%e 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.

%p # Number of ways to write n as a sum of k distinct primes, the smallest

%p # being smalp

%p sumkprims := proc(n,k,smalp)

%p option remember;

%p local a,res,pn;

%p res := n-smalp ;

%p if res < 0 then

%p return 0;

%p elif res > 0 and k <=0 then

%p return 0;

%p elif res = 0 and k = 1 then

%p return 1;

%p else

%p pn := nextprime(smalp) ;

%p a := 0 ;

%p while pn <= res do

%p a := a+procname(res,k-1,pn) ;

%p pn := nextprime(pn) ;

%p end do:

%p a ;

%p end if;

%p end proc:

%p # Number of ways of writing n as a sum of k distinct primes

%p A000586k := proc(n,k)

%p local a,i,smalp ;

%p a := 0 ;

%p for i from 1 do

%p smalp := ithprime(i) ;

%p if k*smalp > n then

%p return a;

%p end if;

%p a := a+sumkprims(n,k,smalp) ;

%p end do:

%p end proc:

%p # Smallest prime which is a sum of n distinct primes

%p A068873 := proc(n)

%p local a,i;

%p a := A007504(n) ;

%p a := nextprime(a-1) ;

%p for i from 1 do

%p if A000586k(a,n) > 0 then

%p return a;

%p end if;

%p a := nextprime(a) ;

%p end do:

%p end proc: # _R. J. Mathar_, May 04 2014

%o (PARI) a(n)=

%o {

%o my(P=primes(n),k=n,t);

%o while(1,

%o forvec(v=vector(n-1,i,[1,k-1]),

%o t=sum(i=1,n-1,P[v[i]])+P[k];

%o if(isprime(t),return(t))

%o ,

%o 2 \\ flag: only strictly increasing vectors v

%o );

%o P=concat(P,nextprime(P[k]+1));

%o k++

%o );

%o } \\ _Charles R Greathouse IV_, Sep 19 2015

%Y Cf. A102330, A013918, A007504.

%K nonn

%O 1,1

%A _Amarnath Murthy_, Mar 19 2002

%E More terms from _Sascha Kurz_, Feb 03 2003

%E Corrected by _Ray Chandler_, Feb 02 2005