A176070 Numbers of the form k^3+k^2+k+1 that are the product of two distinct primes.

15, 85, 259, 1111, 4369, 47989, 65641, 291919, 2016379, 2214031, 3397651, 3820909, 5864581, 9305311, 13881841, 15687751, 16843009, 19756171, 22030681, 28746559, 62256349, 64160401, 74264821, 79692331, 101412319, 117889591, 172189309, 185518471, 191435329, 270004099, 328985791

Offset: 1

Author

Vladimir Joseph Stephan Orlovsky, Apr 07 2010

Keywords

• nonn

Comments

As k^3 + k^2 + k + 1 = (k + 1) * (k^2 + 1) and k <= 1 does not give a term, k + 1 and k^2 + 1 must be prime so k must be even. - David A. Corneth, May 30 2023

Examples

			15 is in the sequence as 15 = 3*5 = 2^3+2^2+2+1; 15 is a product of two distinct primes and of the form k^3 + k^2 + k + 1.
		

Crossrefs

Cf. A002496, A006093, A006881, A053698, A070689, A174969, A176069, A237627 (semiprimes of that form).

Programs

Mathematica

f[n_]:=Last/@FactorInteger[n]=={1,1};Select[Array[ #^3+#^2+#+1&,7! ],f[ # ]&]

PARI

upto(n) = {my(res = List(), u = sqrtnint(n, 3) + 1); forprime(p = 3, u, c = (p-1)^2 + 1; if(isprime(c), listput(res, c*p))); res} \\ David A. Corneth, May 30 2023

Formula

a(n) = (A070689(n + 1) + 1) * (A070689(n + 1)^2 + 1). - David A. Corneth, May 30 2023

Extensions

Name corrected by and more terms from David A. Corneth, May 30 2023