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

1,1

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.

Links

Table of n, a(n) for n=1..50.

Formula

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

Example

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.

Sequence in context: A217776 A039540 A235284 * A250765 A048889 A376539

Adjacent sequences: A373206 A373207 A373208 * A373210 A373211 A373212

Keyword

nonn

Author

Jon E. Schoenfield, Jun 21 2024

Status

approved