A263458 -id:A263458 - cp's OEIS frontend

A374913 Numbers k such that k^(k + 1) == k + 1 (mod 2*k + 1).

2, 3, 6, 11, 14, 15, 18, 23, 26, 30, 35, 39, 50, 51, 54, 63, 74, 75, 78, 83, 86, 90, 95, 98, 99, 111, 114, 119, 131, 134, 135, 138, 146, 155, 158, 174, 179, 183, 186, 191, 194, 198, 210, 215, 219, 230, 231, 239, 243, 251, 254, 270, 278, 299, 303, 306, 315, 323, 326, 330, 338, 350

Offset: 1

Juri-Stepan Gerasimov, Jul 23 2024

nonn

Michel Marcus, Table of n, a(n) for n = 1..10000

Supersequence of A002515 and A374914.

Cf. A374912.

[n: n in [0..350] | n^(n+1) mod (2*n+1) eq n+1];

Select[Range[350],Mod[#^(#+1),2#+1]==#+1 &] (* Stefano Spezia, Jul 23 2024 *)

isok(k) = Mod(k, 2*k+1)^(k+1) == k+1; \\ Michel Marcus, Feb 05 2025

Conjecture (Superseeker): a(n) = A263458(n)/2. - R. J. Mathar, Aug 02 2024

The conjectured formula is false. There exist numbers k such that 2*k + 1 is composite and k^(k + 1) == k + 1 (mod 2*k + 1). For example, when k = 1023: 1023^1024 == 1024 (mod 2047) and 2047 = 23*89 is composite. - Jedrzej Miarecki, Jan 16 2025

A373416 Fixed points of A323712: sizes k of card decks that return to original order after k pile shuffles (but not earlier).

Original entry on oeis.org

1, 3, 4, 6, 9, 12, 22, 27, 28, 36, 46, 52, 60, 70, 78, 81, 100, 102, 148, 166, 172, 180, 190, 196, 198, 238, 243, 262, 268, 270, 292, 310, 316, 348, 358, 366, 372, 382, 388, 420, 460, 462, 478, 486, 502, 508, 540, 556, 598, 606, 612, 646, 652, 660, 676, 700, 708, 718, 729, 742

Offset: 1

R. J. Mathar, Aug 02 2024

nonn

The pile shuffle of a card stack [1,2,3...,n] puts 1 at the bottom of a first auxiliary stack, 2 at the bottom of a second auxiliary stack, 3 above the 1, 4 above the 2, 5 above the 3, 6 above the 4.. eventually the shuffled stack is the first auxiliary stack on top of the second auxiliary stack.

Sequence A263458 contains sizes 2k of card decks that return to reverse order after k pile shuffles. They return to original order after 2k shuffles (2 reversals), so entries of A263458 are candidates for entries here. The case of 30 cards shows that this is not that simple: 30 cards are in reverse order after 5 shuffles, back to original order after 10 shuffles, in reverse order after 15 shuffles (so 30 is in A263458) etc. Because the return to normal order happens already after 10 shuffles (not only at 30), 30 is not in this sequence here.

Cf. A323712, A163781 (even terms halved ?), A263458.

Equals the ordered set {k: A323712(k) = k}.

cp's OEIS Frontend

A374913 Numbers k such that k^(k + 1) == k + 1 (mod 2*k + 1).

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