A383927 Binary echo numbers: positive integers k such that the gpf(k-1) is a suffix of k when gpf(k-1) and k are written in binary.

7, 15, 19, 21, 55, 61, 63, 71, 101, 115, 127, 155, 157, 163, 181, 255, 273, 295, 301, 331, 349, 351, 365, 487, 501, 541, 573, 585, 599, 631, 687, 711, 723, 741, 781, 817, 827, 901, 1055, 1135, 1211, 1277, 1331, 1361, 1387, 1405, 1459, 1471, 1475, 1501, 1621, 1641, 1751

Offset: 1

Author

Michael S. Branicky, May 15 2025

Keywords

• nonn
• base

Comments

No term may be even, since if k were even, then k-1 would be odd and have only odd prime factors, none of which could be a suffix of k.

Examples

			7 is a term since 7 = 111_2, the gpf(6) = 3 = 11_2, and 11 is a suffix of 111.
21 is a term since 21 = 10101_2, the gpf(20) = 5 = 101_2, and 101 is a suffix of 10101.
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..3158

Crossrefs

Binary analog of A383896 (and of A383296).

Cf. A006530.

Programs

Mathematica

Select[Range@2000,(f=IntegerDigits[FactorInteger[#-1][[-1,1]],2])==IntegerDigits[#,2][[-Length@f;;]]&] (* Giorgos Kalogeropoulos, May 15 2025 *)

Python

from sympy import factorint
def ok(n): return n > 2 and bin(n)[2:].endswith(bin(max(factorint(n-1)))[2:])
print([k for k in range(1800) if ok(k)]) # Michael S. Branicky, May 15 2025