A241607 Semiprimes generated by the polynomial (1/4)*(n^5 - 133*n^4 + 6729*n^3 - 158379*n^2 + 1720294*n - 6823316).

5141923, 6084557, 11403823, 13201987, 17488411, 20017609, 33239291, 37446979, 42070423, 47139347, 72512623, 88747907, 118408673, 129881707, 169708339, 184952323, 201267887, 278376073, 324881567, 406044923, 436421497, 538566199, 616639427, 658920007, 750410069

Offset: 1

Author

K. D. Bajpai, Apr 26 2014

Keywords

• nonn

Comments

(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.

For n = 57, this polynomial yields the first semiprime: 5141923 = 821 * 6263.

Examples

			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.
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.
		

Links

• K. D. Bajpai, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A007641 (for primes).

Cf. A001358, A072381, A082919, A121887, A145292, A228183, A237627.

Programs

Maple

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);

Mathematica

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
(*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}]

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