A354846 a(n) is the first composite k such that exactly n primes are the sum of all but one of the numbers from 1 to k-1 that are coprime to k, or -1 if there is no such k.

4, 8, 15, 10, 18, 22, 34, 42, 39, 64, 60, 66, 74, 82, 75, 115, 102, 136, 106, 156, 162, 160, 203, 190, 186, 210, 213, 268, 226, 246, 240, 291, 304, 300, 306, 312, 364, 330, 344, 342, 362, 368, 386, 412, 448, 420, 466, 450, 472, 474, 496, 518, 495, 539, 483, 510, 594, 660, 564, 609, 655, 708, 636

Offset: 1

Author

J. M. Bergot and Robert Israel, Jun 08 2022

Keywords

• nonn

Comments

Conjecture: for every composite k there is at least one such prime.

Examples

			a(3) = 15 because 15 is composite, the numbers from 1 to 14 coprime to 15 are 1, 2, 4, 7, 8, 11, 13, 14, and the 3 primes 47 = 1+2+4+7+8+11+14, 53 = 1+2+4+8+11+13+14 and 59 = 2+4+7+8+11+13+14 are sums of all but one of these.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..1000

Crossrefs

Cf.A000010, A023896, A038566.

Programs

Maple

f:= proc(n) local C,s;
      C:= select(t -> igcd(t,n)=1, [$1..n-1]);
      s:= convert(C,`+`);
      nops(select(isprime,map(t -> s-t, C)))
end proc:
N:= 100; # for a(1)..a(N)
V:= Vector(N): count:= 0:
for nn from 4 while count < N do
    if isprime(nn) then next fi;
    v:= f(nn);
  if v > N then next fi;
  if V[v] = 0 then count:= count+1; V[v]:= nn fi
od:
convert(V,list);