A297849 Composites c where another composite d < c exists such that c and d satisfy c^(d-1) == 1 (mod d^2) and d^(c-1) == 1 (mod c) or satisfy c^(d-1) == 1 (mod d) and d^(c-1) == 1 (mod c^2).

65, 145, 217, 325, 485, 561, 721, 785, 901, 904, 1025, 1105, 1157, 1261, 1281, 1333, 1445, 1729, 1765, 1905, 1937, 2117, 2305, 2465, 2501, 2701, 2705, 3126, 3201, 3365, 3421, 3565, 3601, 3845, 4033, 4097, 4369, 4625, 4901, 5185, 5777, 5833, 6085, 6401, 6499

Offset

1,1

Comments

Are there any composites c where a composite d with d < c exists such that both c^(d-1) == 1 (mod d^2) and d^(c-1) == 1 (mod c^2)?

Links

Table of n, a(n) for n=1..45.

Example

The composites 8 and 65 satisfy the congruences 65^(8-1) == 1 (mod 8^2) and 8^(65-1) == 1 (mod 65), so 65 is a term of the sequence.

Mathematica

With[{s = Select[Range@ 3000, CompositeQ]}, Select[s, Function[c, AnyTrue[Take[s, First@ FirstPosition[s, c]], Or[And[PowerMod[c, (# - 1), #^2] == 1, PowerMod[#, (c - 1), c] == 1], And[PowerMod[c, (# - 1), #] == 1, PowerMod[#, (c - 1), c^2] == 1]] &]]]] (* Michael De Vlieger, Jan 11 2018 *)

Prog

(PARI) is(n) = forcomposite(c=1, n-1, if((Mod(n, c^2)^(c-1)==1 && Mod(c, n)^(n-1)==1) || (Mod(n, c)^(c-1)==1 && Mod(c, n^2)^(n-1)==1), return(1))); 0
forcomposite(c=1, , if(is(c), print1(c, ", ")))

Crossrefs

Subsequence of A270574.

Sequence in context: A194002 A092226 A211255 * A226926 A345700 A121944

Adjacent sequences: A297846 A297847 A297848 * A297850 A297851 A297852

Keyword

nonn

Author

Felix Fröhlich, Jan 07 2018

Status

approved