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

1,1

Comments

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

Links

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

Example

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.

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);

Crossrefs

Cf.A000010, A023896, A038566.

Sequence in context: A076343 A335382 A272346 * A214440 A104101 A130826

Adjacent sequences: A354843 A354844 A354845 * A354847 A354848 A354849

Keyword

nonn

Author

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

Status

approved