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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. 1
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified February 7 17:17 EST 2023. Contains 360128 sequences. (Running on oeis4.)