A056025 Numbers k such that k^12 == 1 (mod 13^2).

1, 19, 22, 23, 70, 80, 89, 99, 146, 147, 150, 168, 170, 188, 191, 192, 239, 249, 258, 268, 315, 316, 319, 337, 339, 357, 360, 361, 408, 418, 427, 437, 484, 485, 488, 506, 508, 526, 529, 530, 577, 587, 596, 606, 653, 654, 657, 675, 677, 695, 698, 699, 746

Offset: 1

Author

Robert G. Wilson v, Jun 08 2000

Keywords

• nonn
• easy

Comments

From 19 to 168 inclusive, these are the numbers that 'fool' the strong pseudoprimality test described in Wilf (1986) in regard to determining whether 169 is composite. - Alonso del Arte, Feb 05 2012

References

• Herbert S. Wilf, Algorithms and Complexity, Englewood Cliffs, New Jersey: Prentice-Hall, 1986, pp. 158-160.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,0,0,0,1,-1).

Crossrefs

Cf. A056021, A056022, A056024, A056026, A056027, A056028, A056031, A056034, A056035.

Cf. A381319 (general case mod n^2).

Programs

Mathematica

Select[ Range[ 800 ], PowerMod[ #, 12, 169 ]==1& ]
LinearRecurrence[{1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1}, {1, 19, 22, 23, 70, 80, 89, 99, 146, 147, 150, 168, 170}, 56] (* Mike Sheppard, Feb 19 2025 *)

PARI

is(k)=Mod(k,169)^12==1 \\ Charles R Greathouse IV, Feb 07 2018

Formula

From Mike Sheppard, Feb 19 2025 : (Start)

a(n) = a(n-1) + a(n-12) - a(n-13).

a(n) = a(n-12) + 13^2.

a(n) ~ (13^2/12)*n. (End)

Extensions

Definition corrected by T. D. Noe, Aug 23 2008