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%I #16 Apr 15 2021 21:39:36
%S 3,71,0,419,71,0,5,11,0,10271,24977,0,29,6869,0,3,9011,0,881,29,0,641,
%T 17,0,41,107,0,17,179,0,5,2801,0,10859,11,0,59,40637,0,461,17957,0,
%U 431,431,0,24977,5,0,12611,599,0,9431,1091,0,107,5867,0,3,15731,0,5,659,0
%N 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.
%C Note that either P or P+2 is composite whenever n is a multiple of 3 and in this case a(n)=0.
%C 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
%H Robert Israel, <a href="/A093245/b093245.txt">Table of n, a(n) for n = 1..10000</a>
%e a(5) = 71: 71 and 73 are twin primes. (71^2+5)^2+5 = 25462121. 25462121 and 25462123 are also twin primes.
%p T:= [3, op(select(t -> isprime(t) and isprime(t+2), [seq(i,i=5..10^7,6)]))]:
%p f:= proc(n) local t,p;
%p if n mod 3 = 0 then return 0 fi;
%p for t in T do
%p p:= (t^2+n)^2+n;
%p if isprime(p) and isprime(p+2) then return t fi
%p od;
%p FAIL
%p end proc:
%p map(f, [$1..100]); # _Robert Israel_, Apr 15 2021
%t 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 *)
%Y Cf. A093189.
%K nonn
%O 1,1
%A _Ray G. Opao_, May 11 2004
%E Corrected and extended by _Robert G. Wilson v_, Sep 02 2004
%E Name amended by _Felix Fröhlich_, Apr 15 2021
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