Giorgos Kalogeropoulos - cp's OEIS frontend

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

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

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

Original entry on oeis.org

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

A383485 a(n) = 2(2(n - 1)! + n + 2) (mod n*(n + 2)).

Original entry on oeis.org

1, 4, 3, 12, 5, 16, 0, 20, 31, 24, 11, 28, 0, 32, 49, 36, 17, 40, 0, 44, 67, 48, 0, 52, 54, 56, 85, 60, 29, 64, 0, 68, 70, 72, 109, 76, 0, 80, 121, 84, 41, 88, 0, 92, 139, 96, 0, 100, 102, 104, 157, 108, 0, 112, 114, 116, 175, 120, 59, 124, 0, 128, 130, 132, 199, 136, 0, 140

Offset: 1

Giorgos Kalogeropoulos, Apr 28 2025

nonn

Fixed points are the lesser of twin primes A001359.

Positions of zeros are primes p such that p + 2 is not a prime A067774.

Cf. A073830, A001359, A067774, A005563.

Table[Mod[2(2(n - 1)! + n + 2), n(n + 2)], {n, 68}]

a(A001359(n)) = n.

a(A067774(n)) = 0.

A380714 a(n) = n*(n-1) mod (10^m-1) where m is the number of decimal digits in n.

Original entry on oeis.org

0, 2, 6, 3, 2, 3, 6, 2, 0, 90, 11, 33, 57, 83, 12, 42, 74, 9, 45, 83, 24, 66, 11, 57, 6, 56, 9, 63, 20, 78, 39, 2, 66, 33, 2, 72, 45, 20, 96, 75, 56, 39, 24, 11, 0, 90, 83, 78, 75, 74, 75, 78, 83, 90, 0, 11, 24, 39, 56, 75, 96, 20, 45, 72, 2, 33, 66

Offset: 1

Giorgos Kalogeropoulos, Mar 27 2025

a(n) = 0 if and only if n is a Kaprekar number (A053816).

Giorgos Kalogeropoulos, Table of n, a(n) for n = 1..9999

Cf. A002283, A002378, A053816, A055642.

a:= n-> n*(n-1) mod (10^length(n)-1):
seq(a(n), n=1..67);  # Alois P. Heinz, Mar 27 2025

Table[Mod[n(n-1), 10^IntegerLength@n - 1], {n,67}]

def a(n): return n*(n-1)%(10**len(str(n))-1)
print([a(n) for n in range(1, 68)]) # Michael S. Branicky, Mar 27 2025

a(n) = A002378(n-1) mod A002283(A055642(n)).

A380585 a(n) = floor(n^2 / 10^m) + (n^2 mod 10^m) where m is the number of decimal digits in n.

Original entry on oeis.org

0, 1, 4, 9, 7, 7, 9, 13, 10, 9, 1, 22, 45, 70, 97, 27, 58, 91, 27, 64, 4, 45, 88, 34, 81, 31, 82, 36, 91, 49, 9, 70, 34, 99, 67, 37, 108, 82, 58, 36, 16, 97, 81, 67, 55, 45, 37, 31, 27, 25, 25, 27, 31, 37, 45, 55, 67, 81, 97, 115, 36, 58, 82, 108, 136, 67, 99

Offset: 0

Giorgos Kalogeropoulos, Mar 27 2025

The positive fixed points of this sequence are the Kaprekar numbers (A053816).

The sum of two halves of the decimal expansion of n^2 after having added a leading 0 if that number of digits is odd. - Michel Marcus, Mar 28 2025

Giorgos Kalogeropoulos, Table of n, a(n) for n = 0..10000
Seiichi Manyama, Post 191006
Ivo Zerkov, 22222^2, seqfan post.

Cf. A053816 (fixed points), A055642, A344851, A358072 (similar plot).

a:= n-> (k-> iquo(n^2, k)+(n^2 mod k))(10^length(n)):
seq(a(n), n=0..66);  # Alois P. Heinz, Mar 27 2025

Table[m=IntegerLength@n; Floor[n^2/10^m] + Mod[n^2,10^m], {n,0,66}]

a(n) = my(d=digits(n^2)); if (#d % 2, d = concat(0, d)); my(m=#d/2); fromdigits(Vec(d,m)) + fromdigits(vector(#d-m, i, d[m+i])); /* Michel Marcus, Mar 28 2025 */

def a(n): return (nn:=n**2)//(M:=10**len(str(n))) + nn%M
print([a(n) for n in range(0, 67)]) # Michael S. Branicky, Mar 27 2025

A381924 Multiplicative order of n mod prime(n).

Original entry on oeis.org

1, 2, 4, 3, 5, 12, 16, 6, 11, 28, 30, 9, 40, 21, 46, 13, 29, 60, 33, 7, 24, 13, 41, 88, 48, 100, 34, 106, 54, 7, 63, 26, 136, 23, 74, 75, 39, 9, 166, 86, 178, 5, 95, 192, 196, 99, 105, 222, 113, 228, 29, 34, 120, 250, 256, 262, 67, 270, 46, 8, 47, 292, 153, 155, 312

Offset: 1

Giorgos Kalogeropoulos, Mar 12 2025

nonn

a(n) is the least k such that prime(n) divides n^k-1.

			a(12) = 9 because the multiplicative order of 12 mod prime(12) is 9.

Vincenzo Librandi, Table of n, a(n) for n = 1..3500

Cf. A226295, A014664, A091185 (n^(-1) mod prime(n)).

[Order(n, NthPrime(n)) : n in [1..65]]; // Vincenzo Librandi, Mar 25 2025

Table[MultiplicativeOrder[n,Prime[n]],{n,65}]

a(n) = znorder(Mod(n, prime(n))); \\ Michel Marcus, Mar 12 2025

If n is a primitive root modulo prime(n), a(n) = prime(n) - 1.

A381969 Primes p with the property that PreviousPrime(p) is a substring of p^2.

Original entry on oeis.org

3701, 65442077, 8410957371097

Offset: 1

Giorgos Kalogeropoulos, Mar 11 2025

a(4) <= 39835421121719177570419.

p = 1410901659681109388941762308365764228483 is also a member of this sequence and it is the only known term that is the greater of a twin prime pair.

			3701 is a term because PreviousPrime(3701) = 3697 is a substring of 3701^2 = 13697401.

Carlos Rivera, Coll.20th-017. The square of Prime contains Prevprime, The Prime Puzzles and Problems Connection.

Cf. A052073.

Select[Prime@Range[10^4],StringContainsQ[ToString[#^2],ToString[NextPrime[#,-1]]]&]

A380787 Odd positive integers k whose continued fraction for sqrt(k) has a central term equal to either floor(sqrt(k)) or floor(sqrt(k)) - 1.

Original entry on oeis.org

3, 7, 11, 19, 23, 27, 31, 43, 47, 51, 59, 67, 71, 79, 83, 103, 107, 119, 123, 127, 131, 139, 151, 163, 167, 171, 179, 187, 191, 199, 211, 223, 227, 239, 243, 251, 263, 267, 271, 283, 287, 291, 307, 311, 331, 339, 343, 347, 359, 363, 367, 379, 383, 387, 391

Offset: 1

Giorgos Kalogeropoulos, Feb 03 2025

nonn

Conjecture: All terms are congruent to 3 mod 4 and all primes of this form (A002145) are terms of the sequence.

			71 is a term because the central element of CF(sqrt(71)) = [8; 2, 2, 1, 7, 1, 2, 2, 16] is 7 and floor(sqrt(71)) - 1 = 7.

Robert Israel, Table of n, a(n) for n = 1..8300

Cf. A002145, A010335, A308778.

filter:= proc(n) local L,v;
  if issqr(n) then return false fi;
  L:= map(op, numtheory:-cfrac(sqrt(n),periodic,quotients));
  if nops(L)::even then return false fi;
  v:=L[(1+nops(L))/2]-floor(sqrt(n));
  v = 0 or v = -1
end proc:
select(filter, [seq(i,i=1..500,2); # Robert Israel, Mar 03 2025

Select[2Range@200+1,(l=Last@ContinuedFraction@Sqrt[#]; m=l[[Floor[Length@l/2]]];m==Floor@Sqrt@#||m==Floor@Sqrt@#-1)&]

A373117 Stable numbers on vertical blade (see the Example section for an explanation).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 20, 22, 30, 33, 40, 44, 50, 55, 60, 66, 70, 77, 80, 88, 90, 99, 100, 101, 102, 110, 111, 113, 121, 124, 131, 135, 141, 146, 151, 157, 161, 168, 171, 179, 181, 191, 200, 201, 202, 204, 212, 215, 220, 222, 226, 232, 237, 242, 248, 252, 259, 262, 272, 282, 292, 300, 303, 306, 311

Offset: 1

Eric Angelini and Giorgos Kalogeropoulos, May 25 2024

			We place the k digits of a number in succession in k adjacent square boxes forming a rectangle of base k and height 1. We can only place a vertical blade under this rectangle in two ways: exactly between two boxes [positions (a) and (d) below] or exactly under the middle of a box [position (b) and (c) below].
If the blade is placed in position (a) between the two 1s of 11, the number 11 is stable.
If the blade is placed in (b) exactly under the 0 of 101, the number 101 is stable.
If the blade is placed in (c) exactly under the 1 of 10, the number is stable.
If the blade is placed in (d) between the 0 and the 2 of 102, the number 102 is stable.
For the last two examples, the balance of the number considered can be explained as follows: the distance which separates a digit from the blade comes into play - the more this distance increases, the heavier the digit in question is. So, the digit 1 of 102 weighs 2 in reality (weight*distance = 1*2 = 2). This quantity balances the influence of the 2 of 102 (for which weight*distance = 2*1 = 2 too).
.
.+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
.|   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |
.+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
.|   | 1 | 1 |   | 1 | 0 | 1 |   | 1 | 0 |   | 1 | 0 | 2 |   |
.+---+---a---+---+---+-b-+---+---+-c-+---+---+---+---d---+---+
.|   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |
.+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
.

Eric Angelini, Balanced numbers, personal blog, May 2024.
Hans Havermann, 10000-term b-file.

Cf. A282115, A282151.

A372408 a(n) = smallest composite not occurring earlier having in decimal representation to its predecessor Levenshtein distance = 1; a(1)=1.

Original entry on oeis.org

1, 4, 6, 8, 9, 39, 30, 10, 12, 14, 15, 16, 18, 28, 20, 21, 22, 24, 25, 26, 27, 57, 50, 40, 42, 32, 33, 34, 35, 36, 38, 48, 44, 45, 46, 49, 69, 60, 62, 52, 51, 54, 55, 56, 58, 68, 63, 64, 65, 66, 76, 70, 72, 74, 75, 77, 78, 88, 80, 81, 82, 84, 85, 86, 87, 187, 117, 110, 100, 102, 104, 105, 106, 108, 118, 111, 112

Offset: 1

Eric Angelini and Giorgos Kalogeropoulos, Apr 29 2024

nonn

The sequence is a permutation of the nonprimes.

			The Levenshtein distance = 1 between 1 and 4, 4 and 6, 6 and 8, 8 and 9, 9 and 39, 39 and 30, 30 and 10, etc.
No smaller composite than 39 was possible for a(6).

Eric Angelini, Prime combination lock, Personal blog, April 2024.

Cf. A000040, A118763, A372407.

a[1]=1;a[n_]:=a[n]=(k=2;While[PrimeQ@k||MemberQ[Array[a,n-1],k]|| EditDistance[ToString@k,ToString@a[n-1]]!=1,k++];k);Array[a,77]

from sympy import isprime
from itertools import islice
from Levenshtein import distance as Ld
def agen(): # generator of terms
    an, aset, mink = 1, {1}, 4
    while True:
        yield an
        s, k = str(an), mink
        while k in aset or Ld(s, str(k)) != 1 or isprime(k): k += 1
        an = k
        aset.add(k)
        while mink in aset or isprime(mink): mink += 1
print(list(islice(agen(), 80))) # Michael S. Branicky, Apr 29 2024

cp's OEIS Frontend

User: Giorgos Kalogeropoulos

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

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

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A383485 a(n) = 2*(2*(n - 1)! + n + 2) (mod n*(n + 2)).

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A380714 a(n) = n*(n-1) mod (10^m-1) where m is the number of decimal digits in n.

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A380585 a(n) = floor(n^2 / 10^m) + (n^2 mod 10^m) where m is the number of decimal digits in n.

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A381924 Multiplicative order of n mod prime(n).

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A381969 Primes p with the property that PreviousPrime(p) is a substring of p^2.

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A383485 a(n) = 2(2(n - 1)! + n + 2) (mod n*(n + 2)).