A383896 -id:A383896 - cp's OEIS frontend

A383296 Primorial base echo numbers: primorial base expansion of k has the largest prime factor of k-1 as its suffix.

11, 41, 59, 101, 127, 137, 161, 221, 229, 251, 313, 323, 337, 401, 551, 641, 667, 703, 757, 799, 881, 929, 1001, 1013, 1093, 1157, 1177, 1211, 1369, 1541, 1583, 1601, 1667, 1753, 1873, 1939, 2017, 2177, 2201, 2393, 2501, 2509, 2561, 2647, 2669, 3043, 3079, 3197, 3217, 3433, 3521, 3613, 3649, 3653, 3823, 3851, 4001

Offset: 1

Antti Karttunen, May 15 2025

			The first seven terms:
    k     A049345(k)  A006530(k-1) (in primorial base)
   11  =        121         5        (21)
   41  =       1121         5        (21)
   59  =       1421        29        (421)
  101  =       3121         5        (21)
  127  =       4101         7        (101)
  137  =       4221        17        (221)
  161  =       5121         5        (21).

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

Primorial base (A049345) analog of A383896.
Subsequence of A383297.
Cf. A006530.

A006530(n) = if(1==n, n, my(f=factor(n)); f[#f~, 1]);
isA383296(n) =  if(n<2, 0, my(p=2, k=A006530(n-1)); while(k, if((k%p)!=(n%p), return(0)); n = n\p; k = k\p; p = nextprime(1+p)); (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

A383907 Echo primes: primes p such that the greatest prime factor of p-1 is a suffix of p.

Original entry on oeis.org

13, 73, 127, 163, 193, 197, 313, 337, 419, 433, 757, 929, 1153, 2017, 2311, 2593, 2647, 3137, 3659, 4483, 4673, 5741, 6857, 7057, 12071, 12097, 13267, 13313, 13619, 14407, 15877, 17191, 18041, 18433, 18439, 19273, 19531, 20353, 21319, 21961, 22279, 24103, 24697, 25411

Offset: 1

Giorgos Kalogeropoulos, May 15 2025

These are primes in A383896(n).
The first pair of twin echo primes is 6467806769 and 6467806771.
Conjecture: There are infinitely many echo primes.

Cf. A000040, A006530, A383896.

Select[Prime@Range@3000,(f=FactorInteger[#-1][[-1,1]]; Mod[#,10^IntegerLength@f]==f)&]

from sympy import factorint, isprime
def ok(n): return n > 2 and isprime(n) and str(n).endswith(str(max(factorint(n-1))))
print([k for k in range(30000) if ok(k)]) # Michael S. Branicky, May 15 2025

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A383296 Primorial base echo numbers: primorial base expansion of k has the largest prime factor of k-1 as its suffix.

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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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A383907 Echo primes: primes p such that the greatest prime factor of p-1 is a suffix of p.

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