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%I #23 Oct 18 2022 03:25:18
%S 5141923,6084557,11403823,13201987,17488411,20017609,33239291,
%T 37446979,42070423,47139347,72512623,88747907,118408673,129881707,
%U 169708339,184952323,201267887,278376073,324881567,406044923,436421497,538566199,616639427,658920007,750410069
%N Semiprimes generated by the polynomial (1/4)*(n^5 - 133*n^4 + 6729*n^3 - 158379*n^2 + 1720294*n - 6823316).
%C (1/4)*(n^5 - 133*n^4 + 6729*n^3 - 158379*n^2 + 1720294*n - 6823316) is a well known prime producing polynomial found by Shyam Sunder Gupta, which generates 57 distinct primes for n = 0,1,...,55,56.
%C For n = 57, this polynomial yields the first semiprime: 5141923 = 821 * 6263.
%H K. D. Bajpai, <a href="/A241607/b241607.txt">Table of n, a(n) for n = 1..10000</a>
%e For n=57: (1/4)*(n^5 - 133*n^4 + 6729*n^3 - 158379*n^2 + 1720294*n - 6823316) = 5141923 = 821 * 6263, which is a semiprime and is included in the sequence.
%e For n=58: (1/4)*(n^5 - 133*n^4 + 6729*n^3 - 158379*n^2 + 1720294*n - 6823316) = 6084557 = 131 * 46447, which is a semiprime and is included in the sequence.
%p with(numtheory): KD:= proc() local a,b,k; k:=(1/4)*(n^5 - 133*n^4 + 6729*n^3 - 158379*n^2 + 1720294*n - 6823316); a:=bigomega(k); if a=2 then RETURN (k); fi; end: seq(KD(), n=0..200);
%t A241607 = {}; Do[k= (1/4) * (n^5 - 133 * n^4 + 6729 * n^3 - 158379 * n^2 + 1720294 * n - 6823316); If[PrimeOmega[k] ==2, AppendTo[A241607, k]], {n,200}]; A241607
%t (*For the b-file:*) n=0;Do[t=((1/4) * (k^5 - 133 * k^4 + 6729 * k^3 - 158379 * k^2 + 1720294 * k - 6823316));If[PrimeOmega[t]==2, n++; Print[n," ",t]], {k,10^6}]
%o (PARI) s=[]; for(n=1, 200, t=(1/4)*(n^5-133*n^4+6729*n^3-158379*n^2+1720294*n-6823316); if(bigomega(t)==2, s=concat(s, t))); s \\ _Colin Barker_, Apr 26 2014
%Y Cf. A007641 (for primes).
%Y Cf. A001358, A072381, A082919, A121887, A145292, A228183, A237627.
%K nonn
%O 1,1
%A _K. D. Bajpai_, Apr 26 2014
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