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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.
2
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
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
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