A338129 Positive numbers k such that the binary representation of k^k ends with that of k.

1, 3, 5, 7, 9, 13, 15, 17, 25, 31, 33, 41, 49, 57, 63, 65, 81, 97, 113, 127, 129, 145, 161, 177, 193, 209, 225, 241, 255, 257, 289, 321, 353, 385, 417, 449, 481, 511, 513, 545, 577, 609, 641, 673, 705, 737, 769, 801, 833, 865, 897, 929, 961, 993, 1023, 1025

Offset: 1

Author

Rémy Sigrist, Oct 11 2020

Keywords

• nonn
• base

Comments

This sequence is infinite as it contains the positive terms of A000225.

All terms are odd.

Run lengths in first differences appear to be regular and suggest a simple procedure to generate the sequence.

Examples

			The binary representation of 3^3 ("11011") ends with that of 3 ("11"), so 3 is a term.
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 1..10000
• Rémy Sigrist, Binary plot of the terms < 2^16
• Index entries for sequences related to automorphic numbers

Crossrefs

Cf. A000225, A082576, A338128, A338130.

Programs

Mathematica

Select[Range[1200],Take[IntegerDigits[#^#,2],-IntegerLength[ #,2]] == IntegerDigits[ #,2]&] (* Harvey P. Dale, Jan 12 2022 *)

PARI

is(n, base=2) = Mod(n, base^#digits(n, base))^n==n