A382617 Numbers k such that k = m*(m^2 + 1) where m^2 + 1 is prime.
2, 10, 68, 222, 1010, 2758, 4112, 8020, 13848, 17602, 46692, 64040, 157518, 175672, 287562, 405298, 592788, 729090, 830678, 1331110, 1561012, 1728120, 1906748, 2000502, 2197130, 2406238, 3112282, 3375150, 3796572, 4096160, 4913170, 5451952, 5832180, 6229688
Offset: 1
Keywords
Examples
For a(1), m = 1, p = 1^2 + 1 = 2, and k = 1*2 = 2. For a(2), m = 2, p = 2^2 + 1 = 5, and k = 2*5 = 10. For a(3), m = 4, p = 4^2 + 1 = 17 and k = 4*17 = 68.
Links
- Alois P. Heinz, Table of n, a(n) for n = 1..10000
- Jon Grantham and Hester Graves, Primes of the Form m^2 + 1 and Goldbach's "Other Other" Conjecture, arXiv preprint arXiv:2502.03513 [math.NT], 2025.
Programs
-
Maple
f:= m-> (p-> `if`(isprime(p), p*m, [][]))(m^2+1): map(f, [$0..200])[]; # Alois P. Heinz, Mar 24 2025
-
Mathematica
Table[m*(m^2 + 1), {m, Select[Range[200], PrimeQ[#^2 + 1] &]}] (* Amiram Eldar, Mar 24 2025 *) -
PARI
apply(x->x*(x^2+1), select(x->isprime(x^2+1), [1..200])) \\ Michel Marcus, Mar 24 2025
-
Python
from sympy import isprime from itertools import count, islice def agen(): # generator of terms yield from (m*p for m in count(1) if isprime(p:=m**2+1)) print(list(islice(agen(), 34))) # Michael S. Branicky, Mar 24 2025
Comments