A144255 Semiprimes of the form k^2+1.

10, 26, 65, 82, 122, 145, 226, 362, 485, 626, 785, 842, 901, 1157, 1226, 1522, 1765, 1937, 2026, 2117, 2305, 2402, 2501, 2602, 2705, 3365, 3482, 3601, 3722, 3845, 4097, 4226, 4762, 5042, 5777, 6085, 6242, 6401, 7226, 7397, 7745, 8465, 9026, 9217, 10001, 10202

Offset: 1

Author

T. D. Noe, Sep 16 2008

Keywords

• nonn
• easy

Comments

Iwaniec proves that there are an infinite number of semiprimes or primes of the form n^2+1. Because n^2+1 is not a square for n>0, all such semiprimes have two distinct prime factors.

Moreover, this implies that one prime factor p of n^2+1 is strictly smaller than n, and therefore also divisor of (the usually much smaller) m^2+1, where m = n % p (binary "mod" operation). - M. F. Hasler, Mar 11 2012

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
• Henryk Iwaniec, Almost-primes represented by quadratic polynomials, Inventiones Mathematicae 47 (2) (1978), pp. 171-188.

Crossrefs

Cf. A001358, A085722, A069987, A193432.

Subsequence of A134406.

Programs

Magma

IsSemiprime:= func; [s: n in [1..100] | IsSemiprime(s) where s is n^2 + 1]; // Vincenzo Librandi, Sep 22 2012

Mathematica

Select[Table[n^2  + 1, {n, 100}], PrimeOmega[#] == 2&] (* Vincenzo Librandi, Sep 22 2012 *)

PARI

select(n->bigomega(n)==2,vector(500,n,n^2+1)) \\ Zak Seidov Feb 24 2011

Python

from sympy import primeomega
from itertools import count, takewhile
def aupto(limit):
    form = takewhile(lambda x: x <= limit, (k**2+1 for k in count(1)))
    return [number for number in form if primeomega(number)==2]
print(aupto(10202)) # Michael S. Branicky, Oct 26 2021

Formula

a(n) = A085722(n)^2 + 1.

Equals { n^2+1 | A193432(n)=2 }. - M. F. Hasler, Mar 11 2012