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

1,1

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

Links

Table of n, a(n) for n=1..31.

Formula

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

Example

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.

Mathematica

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

Prog

(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

Crossrefs

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

Sequence in context: A206169 A160599 A091286 * A160747 A064058 A138322

Adjacent sequences: A176067 A176068 A176069 * A176071 A176072 A176073

Keyword

nonn

Author

Vladimir Joseph Stephan Orlovsky, Apr 07 2010

Extensions

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

Status

approved