A346501 With p = prime(n), a(n) is the least composite k such that A001414(k) = p and k+p is prime, or 0 if there is no such k.

0, 0, 6, 10, 48, 90, 210, 34, 416, 570, 58, 2652, 930, 1984, 1184, 1376, 2820, 118, 1062, 1830, 3660, 4020, 2190, 2370, 1602, 5340, 9006, 12702, 6208, 3090, 8502, 12198, 3810, 7620, 4448, 298, 21372, 17880, 4710, 15386, 7014, 21376, 22836, 11584, 11946, 394, 16548, 40596, 13926, 454, 7136, 6870

Offset

1,3

Comments

All terms are even.

Conjecture: a(n) > 0 for n >= 3.

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Example

a(5) = 48 because A001414(48) = 11 = prime(5) and 48+11 = 59 is prime, and 48 is the least composite that works for this prime.

Maple

spf:= proc(n) local t;
  add(t[1]*t[2], t=ifactors(n)[2])
end proc:
N:= 500: # for a(1)..a(N)
count:= 2:
V:= Vector(N):
for k from 4 by 2 while count < N do
  if isprime(k) then next fi;
  p:= spf(k);
  if isprime(p) then
    m:= numtheory:-pi(p);
    if m <= N and V[m] = 0 and isprime(p+k) then
        V[m]:= k; count:= count+1;
    fi
  fi
od:
convert(V, list);

Mathematica

sopfr[1] = 0; sopfr[n_] := Plus @@ Times @@@ FactorInteger[n]; seq[max_] := Module[{s = Table[0, {max}], c = 2, k = 3, p, ip}, While[c < max, k++; If[CompositeQ[k] && PrimeQ[(p = sopfr[k])] && PrimeQ[k + p] && (ip = PrimePi[p]) <= max && s[[ip]] == 0, c++; s[[ip]] = k]]; s]; seq[50] (* Amiram Eldar, Jul 22 2021 *)

Prog

(PARI) sopfr(n) = (n=factor(n))[, 1]~*n[, 2]; \\ A001414
a(n) = if (n<=2, return(0)); my(p=prime(n)); forcomposite(k=2, , if ((sopfr(k)==p) && isprime(k+p), return (k))); \\ Michel Marcus, Jul 22 2021

Crossrefs

Cf. A001414.

Sequence in context: A068588 A083333 A032359 * A115917 A275004 A115741

Adjacent sequences: A346498 A346499 A346500 * A346502 A346503 A346504

Keyword

nonn,look

Author

J. M. Bergot and Robert Israel, Jul 20 2021

Status

approved