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

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

1,2

Comments

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.

References

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

Links

Table of n, a(n) for n=1..64.

Henryk Iwaniec, Almost-primes represented by quadratic polynomials, Inventiones Mathematicae 47 (2) (1978), pp. 171-188.

Example

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.

Mathematica

Select[Range[100], PrimeQ[(k = #^2 + 1)] || PrimeOmega[k] == 2 &]  (* Amiram Eldar, Mar 30 2020 *)

Crossrefs

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).

Sequence in context: A207333 A179182 A298303 * A364379 A102799 A371738

Adjacent sequences: A333632 A333633 A333634 * A333636 A333637 A333638

Keyword

nonn

Author

Bernard Schott, Mar 30 2020

Status

approved