

A144255


Semiprimes of the form n^2+1.


13



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
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OFFSET

1,1


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


REFERENCES

Iwaniec, H., "Almost primes represented by quadratic polynomials", Invent. math. 47, 1978, 171188.


LINKS

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000


FORMULA

a(n)=A085722(n)^2+1.
A144255 = { n^2+1  A193432(n)=2 }.  M. F. Hasler, Mar 11 2012


MATHEMATICA

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


PROG

(PARI) select(vector(500, n, n^2+1), n>bigomega(n)==2) \\ Zak Seidov Feb 24 2011
(MAGMA) IsSemiprime:= func<n  &+[d[2]: d in Factorization(n)] eq 2>; [s: n in [1..100]  IsSemiprime(s) where s is n^2 + 1]; // Vincenzo Librandi, Sep 22 2012


CROSSREFS

Cf. A001358, A085722, A069987.
Sequence in context: A209983 A055710 A134420 * A259290 A072379 A005970
Adjacent sequences: A144252 A144253 A144254 * A144256 A144257 A144258


KEYWORD

nonn,easy


AUTHOR

T. D. Noe, Sep 16 2008


STATUS

approved



