A371811 -id:A371811 - cp's OEIS frontend

A372771 Numbers m such that the congruence x^(m+1) == m (mod m+1) is solvable.

1, 2, 4, 6, 8, 9, 10, 12, 14, 16, 18, 20, 22, 24, 25, 26, 28, 30, 32, 33, 34, 36, 38, 40, 42, 44, 46, 48, 49, 50, 52, 54, 56, 57, 58, 60, 62, 64, 66, 68, 70, 72, 73, 74, 76, 78, 80, 81, 82, 84, 86, 88, 90, 92, 94, 96, 98, 100, 102, 104, 105, 106, 108, 110

Offset: 1

Juri-Stepan Gerasimov, May 12 2024

nonn

Includes all positive even numbers. - Robert Israel, Mar 14 2025

			9 is a term because x^(9+1) == 9 (mod 9+1) for x = 3 and x = 7, i.e., 3^10 = 59049 == 9 (mod 10) and 7^10 = 282475249 == 9 (mod 10).

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A014117, A330279, A371811.

select(m -> traperror(NumberTheory:-ModularRoot(m,m+1,m+1))::integer, [$1..200]); # Robert Israel, Mar 14 2025

Select[Range[1, 110], With[{m = #}, AnyTrue[Range[1, m + 1], PowerMod[#, m + 1, m + 1] == m &]] &] (* Robert P. P. McKone, May 14 2024 *)

Terms corrected by Robert P. P. McKone, May 14 2024

A375347 a(n) is the number of nonnegative numbers k < n such that the congruence x^k == k (mod n) is solvable.

Original entry on oeis.org

1, 1, 1, 2, 2, 4, 4, 4, 4, 6, 5, 7, 5, 8, 9, 8, 10, 10, 8, 11, 10, 13, 17, 13, 13, 14, 13, 15, 17, 18, 17, 16, 17, 22, 16, 17, 16, 19, 18, 21, 20, 22, 20, 22, 22, 32, 36, 25, 25, 28, 30, 23, 34, 28, 28, 25, 24, 36, 38, 33, 27, 31, 29, 32, 31, 39, 35, 36, 44, 35, 42, 31, 39, 36, 38

Offset: 1

Juri-Stepan Gerasimov, Aug 12 2024

nonn

			a(1) = 1 because x^0 == 0 (mod 1) is solvable where x: 0, 1, 2, 3, 4,.. A001477;
a(2) = 1 because x^0 == 0 (mod 2) is unsolvable,
                 x^1 == 1 (mod 2) is solvable where x: 1, 3, 5, 7, 9,.. A005408;
a(3) = 1 because x^0 == 0 (mod 3) is unsolvable,
                 x^1 == 1 (mod 3) is solvable where x: 1, 4, 7, 10, 13,.. A016777,
                 x^2 == 2 (mod 3) is unsolvable;
a(4) = 2 because x^0 == 0 (mod 4) is unsolvable,
                 x^1 == 1 (mod 4) is solvable where x: 1, 5, 9, 13, 16,.. A016813,
                 x^2 == 2 (mod 4) is unsolvable,
                 x^3 == 3 (mod 4) is solvable where x: 3, 7, 11, 15, 19,.. A004767.

Cf. A371811, A372771.

Cf. A001477, A004767, A005408, A016777, A016813.

is(k, n) = for (i=0, n-1, if (Mod(i, n)^k == k, return(1)));
a(n) = sum(k=0, n-1, is(k, n)); \\ Michel Marcus, Aug 13 2024

More terms from Michel Marcus, Aug 13 2024

A373899 Semiprimes q*p such that q^p == p (mod (q - p)), where q > p.

Original entry on oeis.org

6, 15, 21, 33, 35, 55, 65, 77, 85, 91, 133, 143, 145, 155, 161, 187, 209, 217, 221, 247, 253, 265, 299, 301, 323, 341, 377, 391, 403, 415, 437, 451, 481, 493, 533, 545, 551, 553, 559, 581, 589, 611, 629, 667, 671, 689, 697, 703, 713, 781, 793, 799, 817, 893, 899, 901

Offset: 1

Juri-Stepan Gerasimov, Jun 22 2024

nonn

			15 = 3*5 is a term because 5^3 == 3 (mod 2).

Subsequence of A001358.

Cf. A037074 (subsequence), A046388, A371811.

seqQ[n_] := Module[{f = FactorInteger[n], p, q}, If[f[[;; , 2]] == {1, 1}, p = f[[1, 1]]; q = f[[2, 1]]; PowerMod[q, p, q - p] == Mod[p, q - p], False]]; Select[Range[1000], seqQ] (* Amiram Eldar, Jun 26 2024 *)

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A372771 Numbers m such that the congruence x^(m+1) == m (mod m+1) is solvable.

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A375347 a(n) is the number of nonnegative numbers k < n such that the congruence x^k == k (mod n) is solvable.

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A373899 Semiprimes q*p such that q^p == p (mod (q - p)), where q > p.

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Mathematica