A102330 -id:A102330 - cp's OEIS frontend

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

Amarnath Murthy, Mar 19 2002

nonn

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

			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.

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.

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).

Cf. A102330, A013918, A007504.

# 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

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

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

More terms from Sascha Kurz, Feb 03 2003

Corrected by Ray Chandler, Feb 02 2005

A383725 a(n) is the least number k such that omega(k) = n and the largest prime factor of k equals the sum of its remaining prime factors, where omega(k) = A001221(k).

Original entry on oeis.org

30, 3135, 3570, 844305, 1231230, 463798335, 1089218130, 410825520105, 905980145070, 818186519485335, 1461885412557570, 2023416377587710105, 3676255934199278430, 6175645531427513476335, 14590719651042312667890, 29263451149172039260325865, 67794672364404337821058590

Offset: 3

Paolo Xausa, May 07 2025

nonn

a(n) and n have opposite parity. - David A. Corneth, May 08 2025

			a(3) = 30 is the smallest number having 3 distinct prime factors (namely 2, 3, and 5) such that the largest one is the sum of the others (2 + 3 = 5).
a(4) = 3135 is the smallest number having 4 distinct prime factors (namely 3, 5, 11 and 19) such that the largest one is the sum of the others (3 + 5 + 11 = 19).

Cf. A001221, A002110, A068873, A102330, A365795, A382469, A383728, A383729.

First column of A383726.

isok(k, n) = my(f=factor(k)); (omega(f)==n) && (vecsum(f[,1]) == 2*vecmax(f[,1]));
a(n) = my(k=1); while (!isok(k, n), k++); k; \\ Michel Marcus, May 08 2025

a(n) = (Product_{i=1..n-1} A102330(n-1,i))*A068873(n-1).
a(n) = A383726(n,1).
a(n) >= A002110(n).

A102325 Primes p such that the largest prime divisor of p^3 + 1 is less than p.

Original entry on oeis.org

17, 19, 23, 31, 101, 103, 173, 179, 257, 263, 293, 353, 373, 431, 467, 491, 521, 563, 593, 619, 641, 677, 719, 739, 773, 821, 829, 857, 859, 863, 881, 929, 941, 953, 1031, 1051, 1087, 1091, 1109, 1129, 1229, 1297, 1327, 1399, 1433, 1487, 1489, 1499, 1583

Offset: 1

Labos Elemer, Jan 05 2005

nonn

			p = 17, 1 + p^3 = 1 + 4913 = 2*3*3*3*7*13, so the largest prime factor is 13 < p = 17.

Giovanni Resta, Table of n, a(n) for n = 1..10000

Cf. A000040, A065091, A073501, A102326-A102330.

Mathematica
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<Ray Chandler, Jan 08 2005 *)
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A068873 Smallest prime which is a sum of n distinct primes.

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A383725 a(n) is the least number k such that omega(k) = n and the largest prime factor of k equals the sum of its remaining prime factors, where omega(k) = A001221(k).

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A102325 Primes p such that the largest prime divisor of p^3 + 1 is less than p.

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Author

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Examples

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Programs

Mathematica