A373209 Numbers k such that k^2 - 1 and k^2 + 1 have 8 divisors each.

68, 112, 128, 162, 200, 212, 252, 294, 318, 336, 338, 372, 448, 450, 498, 502, 542, 578, 592, 598, 612, 648, 672, 678, 708, 752, 762, 808, 812, 852, 878, 888, 938, 952, 992, 996, 1012, 1038, 1098, 1102, 1116, 1122, 1188, 1202, 1212, 1248, 1258, 1328, 1362, 1380

Offset: 1

Author

Jon E. Schoenfield, Jun 21 2024

Keywords

• nonn

Comments

Among the first 10000 terms (from a(1) = 68 through a(10000) = 697578), k^2 - 1 and k^2 + 1 are each the product of three distinct primes, except for

125 terms for which k^2 + 1 = 5^3 times a prime

6 terms for which k^2 + 1 = 13^3 times a prime

1 terms for which k^2 + 1 = 17^3 times a prime

1 terms for which k^2 + 1 = 29^3 times a prime, and

4 terms for which k^2 - 1 = p^3 * (p^3 +/- 2) (with p = 19, 29, 37, 83, respectively).

The first term for which both k^2 - 1 and k^2 + 1 are of the form p^3 * q is k = 41457661182: k^2 - 1 = 3461^3 * 41457661183, while k^2 + 1 = 5^3 * 13749901365452077097.

Examples

			68 is a term: both 68^2 - 1 = 4623 = 3 * 23 * 67 and 68^2 + 1 = 4625 = 5^3 * 37 have 8 divisors.
		

Crossrefs

Cf. A000005, A002522, A005563, A069062, A108278, A193432, A347191.

Formula

{ k : tau(k^2 - 1) = tau(k^2 + 1) = 8}, where tau() is the number of divisors function, A000005.