A259021 Numbers k such that k^2 = Product_{d|k} d (= A007955(k)) and simultaneously k^2 + 1 is a divisorial prime (A258455).

1, 6, 10, 14, 26, 74, 94, 134, 146, 206, 314, 326, 386, 466, 634, 674, 1094, 1174, 1294, 1306, 1354, 1366, 1546, 1654, 1766, 1774, 1894, 1966, 2026, 2126, 2174, 2326, 2594, 2654, 2746, 2974, 2986, 3046, 3106, 3134, 3214, 3254, 3274, 3314, 3326, 3334, 3446

Offset: 1

Author

Jaroslav Krizek, Sep 01 2015

Keywords

• nonn

Comments

First deviation from A259020 is at a(15).

With number 2 complement of A259023 with respect to A118369.

1 together with squarefree semiprimes (A006881) k such that k^2 + 1 is prime. Without the squarefree restriction there will be only one more term, 4. - Amiram Eldar, Sep 25 2022

Examples

			The number 10 is in sequence because 10^2 = 1*2*5*10 = 100 and simultaneously 101 is prime.
		

Links

• Danny Rorabaugh, Table of n, a(n) for n = 1..10000

Crossrefs

Union of {1} and (intersection of A005574 and A006881).

Subsequence of A007422, A048943, A259020, A118369.

Cf. A007955, A007422, A048943, A052291, A118369, A258455, A259020, A259023, A258897.

Programs

Magma

[Floor(Sqrt(n-1)): n in [1..10000000] | IsPrime(n) and n-1 eq (&*(Divisors(Floor(Sqrt(n-1)))))];

Mathematica

Prepend[2*Select[Prime[Range[2, 300]], PrimeQ[4 #^2 + 1] &], 1] (* Amiram Eldar, Sep 25 2022 *)

Sage

a = [n for n in range(1,100000) if is_prime(n^2+1) and n^2==prod(list(divisors(n)))] # Danny Rorabaugh, Sep 21 2015

Formula

a(n) = 2*A052291(n) for n > 1. - Amiram Eldar, Sep 25 2022