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

%I A383896 #39 May 29 2025 00:53:15
%S A383896 13,57,73,111,127,163,193,197,313,323,337,419,433,687,757,817,847,897,
%T A383896 929,931,973,1037,1153,1177,1211,1641,2017,2311,2593,2623,2647,2913,
%U A383896 3073,3137,3659,3661,3829,4031,4117,4213,4453,4483,4537,4673,4737,4971,5377,5741
%N A383896 Echo numbers: positive integers k such that the largest prime factor of k-1 is a suffix of k.
%C A383896 They are called like that because k-1 leaves an echo in the decimal representation of k.
%C A383896 There are infinitely many terms: 56^i+1 is a term for i > 0.
%C A383896 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
%H A383896 Giorgos Kalogeropoulos, <a href="/A383896/b383896.txt">Table of n, a(n) for n = 1..10000</a>
%H A383896 Code Golf Stack Exchange, <a href="https://codegolf.stackexchange.com/questions/279790/output-the-echo-numbers">Output the Echo Numbers</a>
%H A383896 Giorgos Kalogeropoulos, <a href="/A383896/a383896.txt">Table of n, a(n) for n = 1..100000</a>
%e A383896 k = 4971 is an echo number because k-1 = 4970 = 2*5*7*71 and k ends in 71.
%p A383896 filter:= proc(n) local p;
%p A383896   p:= max(numtheory:-factorset(n-1));
%p A383896   n - p mod 10^(1+ilog10(p)) = 0
%p A383896 end proc:
%p A383896 select(filter, [seq(i,i=11..10000,2)]); # _Robert Israel_, May 14 2025
%t A383896 Select[Range[2,6000],(f=FactorInteger[#-1][[-1,1]];Mod[#,10^IntegerLength@f]==f)&]
%o A383896 (Python)
%o A383896 from sympy import factorint
%o A383896 def ok(n): return n > 2 and str(n).endswith(str(max(factorint(n-1))))
%o A383896 print([k for k in range(6000) if ok(k)]) # _Michael S. Branicky_, May 14 2025
%o A383896 (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
%Y A383896 Cf. A006530.
%Y A383896 Cf. A383296 (primorial base analog), A383927 (binary analog).
%K A383896 nonn,base
%O A383896 1,1
%A A383896 _Giorgos Kalogeropoulos_, May 14 2025