A357816 a(n) is the first even number k such that there are exactly n pairs (p,q) where p and q are prime, p<=q, p+q = k, and p+A001414(k) and q+A001414(k) are also prime.

2, 16, 60, 72, 220, 132, 374, 276, 492, 638, 636, 852, 620, 854, 996, 1056, 1026, 1212, 2070, 1530, 2610, 3976, 3844, 1488, 1572, 4812, 4770, 3942, 2484, 5028, 3234, 4668, 6036, 3276, 5172, 5532, 6756, 2730, 6084, 4230, 6390, 9132, 14134, 4620, 9674, 10692, 6600, 8910, 10836, 12204, 18852, 9660

Offset

0,1

Links

Robert Israel, Table of n, a(n) for n = 0..500

Example

a(3) = 72 because A001414(72) = 12 and there are 3 pairs: (5,67), (11,61) and (31,41) where 5+67 = 11+61 = 31+41 = 72 and 5, 5+12 = 17, 67, 67+12 = 79, 11, 11+12 = 23, 61, 61+12 = 73, 31, 31+12 = 43, 41, and 41+12 = 53 are all prime; and this is the first even number with 3 such pairs.

Maple

sp:= proc(n) local t; add(t[1]*t[2], t=ifactors(n)[2]) end proc:
f:= proc(n) local s, p, q, count;
    s:= sp(n);
    if s::odd then return 0 fi;
    p:= 2; count:= 0;
    do
      p:= nextprime(p);
      q:= n-p;
      if p > q then return count fi;
      if isprime(p+s) and isprime(q) and isprime(q+s) then count:= count+1 fi;
    od;
end proc:
V:= Array(0..60): count:= 0:
for n from 2 by 2 while count < 61 do
  v:= f(n);
  if v <= 60 and V[v] = 0 then V[v]:= n; count:= count+1;  fi
od:
convert(V, list);

Mathematica

a[n_] := Block[{k=2, s}, While[True, s = Plus @@ Times @@@ FactorInteger@ k; If[n == Length@ Select[ Prime@ Range@ PrimePi[k/2], And @@ PrimeQ@ {k-#, #+s, k-#+s} &], Break[]]; k += 2]; k]; a /@ Range[0, 20] (* Giovanni Resta, Oct 24 2022 *)

Crossrefs

Cf. A001414, A023036.

Sequence in context: A123791 A293620 A206980 * A170991 A209219 A207688

Adjacent sequences: A357813 A357814 A357815 * A357817 A357818 A357819

Keyword

nonn

Author

J. M. Bergot and Robert Israel, Oct 13 2022

Status

approved