A383296 -id:A383296 - cp's OEIS frontend

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

Giorgos Kalogeropoulos, May 14 2025

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

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

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

Cf. A006530.

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

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

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

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

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

A383297 Numbers k for which A276086(A006530(k-1)) divides A276086(k), where A006530 gives the largest prime factor of k, and A276086 is the primorial base exp-function.

Original entry on oeis.org

3, 5, 9, 11, 15, 17, 23, 27, 29, 33, 41, 43, 53, 57, 59, 63, 65, 71, 75, 79, 83, 85, 87, 89, 99, 101, 105, 113, 115, 119, 123, 125, 127, 129, 135, 137, 141, 143, 147, 149, 161, 169, 173, 179, 187, 195, 197, 203, 207, 209, 221, 225, 229, 235, 239, 249, 251, 253, 257, 259, 261, 267, 281, 287, 293, 295, 297, 311, 313

Offset: 1

Antti Karttunen, May 15 2025

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences related to primorial base.

Cf. A006530, A276086, A383296 (subsequence).

A006530(n) = if(1==n, n, my(f=factor(n)); f[#f~, 1]);
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
isA383297(n) = ((n>1) && !(A276086(n)%A276086(A006530(n-1))));

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.

Original entry on oeis.org

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

Michael S. Branicky, May 15 2025

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.

			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.

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

Binary analog of A383896 (and of A383296).

Cf. A006530.

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

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

cp's OEIS Frontend

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

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A383297 Numbers k for which A276086(A006530(k-1)) divides A276086(k), where A006530 gives the largest prime factor of k, and A276086 is the primorial base exp-function.

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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.

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