A248742 -id:A248742 - cp's OEIS frontend

1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 14, 15, 16, 19, 20, 22, 24, 25, 26, 28, 29, 30, 34, 35, 36, 39, 40, 42, 44, 45, 46, 48, 49, 50, 51, 52, 54, 56, 58, 59, 60, 61, 62, 64, 65, 66, 69, 71, 74, 76, 78, 79, 80, 84, 85, 86, 88, 90, 92, 94, 95, 96, 100

Offset: 1

Bernard Schott, Mar 30 2020

nonn

Equivalently, numbers m such that m^2 + 1 is prime or semiprime.
Henryk Iwaniec proved in 1978 that this sequence is infinite (see link). By contrast, it is not known whether there are infinitely many primes of the form m^2 + 1 (or infinitely many semiprimes of that form).
The integers that have at most 2 prime factors counted with multiplicity are called almost-primes of order 2 and they are in A037143. Here, as m^2 + 1 is not a square for m > 0, all the semiprimes of this form have two distinct prime factors (A144255), and with the primes of the form m^2 + 1 (A002496), they constitute A248742.

			10^2 + 1 = 101, which is prime, so 10 is in the sequence.
11^2 + 1 = 122 = 2 * 61, so 11 is in the sequence.
12^2 + 1 = 145 = 5 * 29, so 12 is in the sequence.
13^2 + 1 = 170 = 2 * 5 * 17, so 13 is not in the sequence.

R. K. Guy, Unsolved Problems in Number Theory, A1.

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Henryk Iwaniec, Almost-primes represented by quadratic polynomials, Inventiones Mathematicae 47 (2) (1978), pp. 171-188.

Union of A005574 and A085722.
Cf. A002496 (m^2 + 1 is prime), A005574 (corresponding m).
Cf. A144255 (m^2 + 1 is semiprime), A085722 (corresponding m).
Cf. A248742 (m^2 + 1 is prime or semiprime), this sequence (corresponding m).
Cf. A037143 (numbers with at most 2 prime factors counted with multiplicity).

Select[Range[100], PrimeQ[(k = #^2 + 1)] || PrimeOmega[k] == 2 &]  (* Amiram Eldar, Mar 30 2020 *)
Select[Range[100],PrimeOmega[#^2+1]<3&] (* Harvey P. Dale, Aug 08 2025 *)

cp's OEIS Frontend

A333635 Numbers m such that m^2 + 1 has at most 2 prime factors.

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