A383896 Echo numbers: positive integers k such that the largest prime factor of k-1 is a suffix of k.

13, 57, 73, 111, 127, 163, 193, 197, 313, 323, 337, 419, 433, 687, 757, 817, 847, 897, 929, 931, 973, 1037, 1153, 1177, 1211, 1641, 2017, 2311, 2593, 2623, 2647, 2913, 3073, 3137, 3659, 3661, 3829, 4031, 4117, 4213, 4453, 4483, 4537, 4673, 4737, 4971, 5377, 5741

Offset: 1

Author

Giorgos Kalogeropoulos, May 14 2025

Keywords

• nonn
• base

Comments

They are called like that because k-1 leaves an echo in the decimal representation of k.

There are infinitely many terms: 56^i+1 is a term for i > 0.

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. - Michael S. Branicky, May 14 2025

Examples

			k = 4971 is an echo number because k-1 = 4970 = 2*5*7*71 and k ends in 71.
		

Links

• Giorgos Kalogeropoulos, Table of n, a(n) for n = 1..10000
• Code Golf Stack Exchange, Output the Echo Numbers
• Giorgos Kalogeropoulos, Table of n, a(n) for n = 1..100000

Crossrefs

Cf. A006530.

Cf. A383296 (primorial base analog), A383927 (binary analog).

Programs

Maple

filter:= proc(n) local p;
  p:= max(numtheory:-factorset(n-1));
  n - p mod 10^(1+ilog10(p)) = 0
end proc:
select(filter, [seq(i,i=11..10000,2)]); # Robert Israel, May 14 2025

Mathematica

Select[Range[2,6000],(f=FactorInteger[#-1][[-1,1]];Mod[#,10^IntegerLength@f]==f)&]

PARI

isok(k) = if (k>2, my(x = vecmax(factor(k-1)[,1]), m = 1+logint(x, 10)); k % 10^m == x); \\ Michel Marcus, May 14 2025

Python

from sympy import factorint
def ok(n): return n > 2 and str(n).endswith(str(max(factorint(n-1))))
print([k for k in range(6000) if ok(k)]) # Michael S. Branicky, May 14 2025