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

1,1

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

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

Example

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

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)&]

Prog

(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
(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

Crossrefs

Cf. A006530.

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

Sequence in context: A212053 A264853 A210290 * A007202 A368813 A222161

Adjacent sequences: A383893 A383894 A383895 * A383897 A383898 A383899

Keyword

nonn,base

Author

Giorgos Kalogeropoulos, May 14 2025

Status

approved