A093245 a(n) is the lesser term of the smallest twin prime pair such that if P=(a(n)^2+n)^2+n, then P and P+2 are also twin primes. a(n) is 0 if no such pair exists.

3, 71, 0, 419, 71, 0, 5, 11, 0, 10271, 24977, 0, 29, 6869, 0, 3, 9011, 0, 881, 29, 0, 641, 17, 0, 41, 107, 0, 17, 179, 0, 5, 2801, 0, 10859, 11, 0, 59, 40637, 0, 461, 17957, 0, 431, 431, 0, 24977, 5, 0, 12611, 599, 0, 9431, 1091, 0, 107, 5867, 0, 3, 15731, 0, 5, 659, 0

Offset

1,1

Comments

Note that either P or P+2 is composite whenever n is a multiple of 3 and in this case a(n)=0.

Conjecture: a(n) = 0 only if n is a multiple of 3. Note that this implies the existence of infinitely many twin primes. - Robert Israel, Apr 15 2021

Links

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

Example

a(5) = 71: 71 and 73 are twin primes. (71^2+5)^2+5 = 25462121. 25462121 and 25462123 are also twin primes.

Maple

T:= [3, op(select(t -> isprime(t) and isprime(t+2), [seq(i, i=5..10^7, 6)]))]:
f:= proc(n) local t, p;
    if n mod 3 = 0 then return 0 fi;
    for t in T do
      p:= (t^2+n)^2+n;
      if isprime(p) and isprime(p+2) then return t fi
    od;
    FAIL
end proc:
map(f, [$1..100]); # Robert Israel, Apr 15 2021

Mathematica

f[n_] := Block[{k = 2}, If[ Mod[n, 3] != 0, While[ p = Prime[k]; q = (p^2 + n)^2 + n; !PrimeQ[p + 2] || !PrimeQ[q] || !PrimeQ[q + 2], k++ ]; p, 0]]; Table[ f[n], {n, 63}] (* Robert G. Wilson v, Sep 02 2004 *)

Crossrefs

Cf. A093189.

Sequence in context: A140048 A135951 A361254 * A108231 A130894 A254665

Adjacent sequences: A093242 A093243 A093244 * A093246 A093247 A093248

Keyword

nonn

Author

Ray G. Opao, May 11 2004

Extensions

Corrected and extended by Robert G. Wilson v, Sep 02 2004

Name amended by Felix Fröhlich, Apr 15 2021

Status

approved