A338128 -id:A338128 - cp's OEIS frontend

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

Rémy Sigrist, Oct 11 2020

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.

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

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

Cf. A000225, A082576, A338128, A338130.

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

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

A338130 Positive numbers k such that the ternary representation of k^k ends with that of k.

Original entry on oeis.org

1, 4, 7, 10, 19, 28, 37, 46, 55, 64, 73, 82, 109, 136, 163, 190, 217, 244, 271, 298, 325, 352, 379, 406, 433, 460, 487, 514, 541, 568, 595, 622, 649, 676, 703, 730, 811, 892, 973, 1054, 1135, 1216, 1297, 1378, 1459, 1540, 1621, 1702, 1783, 1864, 1945, 2026

Offset: 1

Rémy Sigrist, Oct 11 2020

All terms are of the form 3*m + 1 for some m >= 0.
The first differences appear to contain only powers of 3 and to be weakly increasing.
Run lengths in first differences appear to be regular and suggest a simple procedure to generate the sequence.

			The ternary representation of 4^4 ("100111") ends with that of 4 ("11"), so 4 is a term.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, Ternary plot of the terms < 3^10 (blue squares correspond to 1's and red squares to 2's)
Index entries for sequences related to automorphic numbers

Cf. A082576, A338128, A338129.

Select[Range[2100],Take[IntegerDigits[#^#,3],-IntegerLength[#,3]] == IntegerDigits[ #,3]&] (* Harvey P. Dale, Feb 13 2022 *)

is(n, base=3) = Mod(n, base^#digits(n, base))^n==n

cp's OEIS Frontend

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

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