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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.

%I A354846 #11 Jun 14 2022 15:30:36
%S A354846 4,8,15,10,18,22,34,42,39,64,60,66,74,82,75,115,102,136,106,156,162,
%T A354846 160,203,190,186,210,213,268,226,246,240,291,304,300,306,312,364,330,
%U A354846 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
%N 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.
%C A354846 Conjecture: for every composite k there is at least one such prime.
%H A354846 Robert Israel, <a href="/A354846/b354846.txt">Table of n, a(n) for n = 1..1000</a>
%e A354846 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.
%p A354846 f:= proc(n) local C,s;
%p A354846       C:= select(t -> igcd(t,n)=1, [$1..n-1]);
%p A354846       s:= convert(C,`+`);
%p A354846       nops(select(isprime,map(t -> s-t, C)))
%p A354846 end proc:
%p A354846 N:= 100; # for a(1)..a(N)
%p A354846 V:= Vector(N): count:= 0:
%p A354846 for nn from 4 while count < N do
%p A354846     if isprime(nn) then next fi;
%p A354846     v:= f(nn);
%p A354846   if v > N then next fi;
%p A354846   if V[v] = 0 then count:= count+1; V[v]:= nn fi
%p A354846 od:
%p A354846 convert(V,list);
%Y A354846 Cf.A000010, A023896, A038566.
%K A354846 nonn
%O A354846 1,1
%A A354846 _J. M. Bergot_ and _Robert Israel_, Jun 08 2022