A239019 -id:A239019 - cp's OEIS frontend

A317291 Numbers k not in A239019 such that A177894(k) = 0.

0, 1001, 1012, 1023, 1034, 1045, 1056, 1067, 1078, 1089, 1100, 1122, 1133, 1144, 1155, 1166, 1177, 1188, 1199, 1210, 1221, 1232, 1243, 1254, 1265, 1276, 1287, 1298, 1320, 1331, 1342, 1353, 1364, 1375, 1386, 1397, 1430, 1441, 1452, 1463, 1474, 1485, 1496

Offset: 1

Jianing Song, Jan 21 2019

Of course A177894(A239019(n)) = 0 because the corresponding circular matrices each contains at least two identical rows or columns. This sequence gives the other numbers.

A four-digit number abcd is in this sequence if and only if a + c = b + d and (a != c or b != d).

			2178 is a term because the value of the following determinant is 0, although the determinant itself contains no identical rows or columns:
| 2 1 7 8 |
| 1 7 8 2 |
| 7 8 2 1 |
| 8 2 1 7 |

Eric Weisstein's World of Mathematics, Circulant Matrix.
Wikipedia, Circulant matrix.

Cf. A177894, A239019, A299792.

for(n=0, 1500, if(!n ||(n&!A177894(n)&&!A239019(n)),print1(n, ", "))) \\ See A177894 and A239019 for their programs

A299792 Numbers k such that A177894(k) = 0.

Original entry on oeis.org

0, 11, 22, 33, 44, 55, 66, 77, 88, 99, 111, 222, 333, 444, 555, 666, 777, 888, 999, 1001, 1010, 1012, 1023, 1034, 1045, 1056, 1067, 1078, 1089, 1100, 1111, 1122, 1133, 1144, 1155, 1166, 1177, 1188, 1199, 1210, 1212, 1221, 1232, 1243, 1254, 1265, 1276, 1287, 1298, 1313

Offset: 1

Jianing Song, Jan 21 2019

A one-digit number a is in this sequence if and only if a = 0.
A two-digit number ab is in this sequence if and only if a = b.
A three-digit number abc is in this sequence if and only if a = b = c.
A four-digit number abcd is in this sequence if and only if a + c = b + d or (a = c and b = d)
A239019 is trivially a subsequence (because the corresponding circular matrices each contains at least two identical rows or columns). {a(n)} \ A239019 is given as A317291.

			1452 is a term because the value of the following determinant is 0:
  | 1 4 5 2 |
  | 4 5 2 1 |
  | 5 2 1 4 |
  | 2 1 4 5 |

Paolo Xausa, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Circulant Matrix
Wikipedia, Circulant matrix

Cf. A177894, A239019, A317291.

A299792Q[k_] := k == 0 || Det[NestList[RotateLeft, IntegerDigits[k], IntegerLength[k]-1]] == 0; Select[Range[0, 2000], A299792Q] (* Paolo Xausa, Mar 11 2024 *)

for(n=0, 1500, if(!A177894(n), print1(n, ", "))) \\ See A177894 for its program

A286138 Pseudo-palindromic numbers: not palindromes (A002113), but a nontrivial palindromic concatenation (AA or ABA) of arbitrary nonzero integers A and B.

Original entry on oeis.org

1010, 1101, 1121, 1131, 1141, 1151, 1161, 1171, 1181, 1191, 1201, 1211, 1212, 1231, 1241, 1251, 1261, 1271, 1281, 1291, 1301, 1311, 1313, 1321, 1341, 1351, 1361, 1371, 1381, 1391, 1401, 1411, 1414, 1421, 1431, 1451, 1461, 1471, 1481, 1491, 1501, 1511, 1515, 1521, 1531

Offset: 1

M. F. Hasler, May 03 2017

The pseudo- or almost-palindromic numbers considered here are not related to the similarly named but different concepts mentioned in comments on A003555 and in A060087 - A060088.
We could consider "more general" palindromic concatenations like A.B.B.A, A.B.C.B.A, etc., but all of these can be written as A.B'.A with B' = B.B resp. B.C.B, etc. The result is non-palindromic (i.e., not in A002113) as required, if and only if at least one of the strings is non-palindromic.
Here, A is allowed to have only one digit, so most of the first 100 terms are of the form 1.B.1 where B = 10, 12, 13, ... (palindromes 11, 22, 33, ... excluded).
If all of the strings A, B (...) are required to be non-palindromic, the sequence starts with terms of the form A.A with A = 10, 12, 13, ..., 98:  1010, 1212, 1313, 1414, 1515, 1616, 1717, 1818, 1919, 2020, 2121, 2323, .... This is a subsequence of A239019 (numbers which are not primitive words over the alphabet {0,...,9} when written in base 10).

A286138 = select(t->!is_A002113(t),setunion(vector(801,i,((i-1)\89+1)*1001+((i-1)%89+1)*10),vector(89,i,(i+9)*101))) \\ The first 810 terms.

A369347 Numbers whose decimal expansion is quasiperiodic.

Original entry on oeis.org

11, 22, 33, 44, 55, 66, 77, 88, 99, 111, 222, 333, 444, 555, 666, 777, 888, 999, 1010, 1111, 1212, 1313, 1414, 1515, 1616, 1717, 1818, 1919, 2020, 2121, 2222, 2323, 2424, 2525, 2626, 2727, 2828, 2929, 3030, 3131, 3232, 3333, 3434, 3535, 3636, 3737, 3838, 3939

Offset: 1

Rémy Sigrist, Jan 21 2024

The decimal representation of a term (ignoring leading zeros) can be covered by (possibly overlapping) occurrences of one of its proper prefixes.
This sequence contains, among others, A020338 and A239019.
The first term that does not belong to A239019 is a(109) = 10101.

			The number 10101101 belongs to this sequence as its decimal expansion can be covered by copies of its proper prefix 101:
      101
        101
           101
      ........
      10101101

Index entries for sequences related to decimal expansion of n

Cf. A020338, A239019, A320441 (binary analog).

is(w) = { my (tt=0); for (l=1, oo, my (t=w%(10^l)); if (t!=tt, if (t==w, return (0)); my (r=w, g=l); while (g-->=0 && r>=t, r \= 10; if (r%(10^l)==t, if (r==t, return (1), g=l))); tt = t)) }

cp's OEIS Frontend

A317291 Numbers k not in A239019 such that A177894(k) = 0.

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A299792 Numbers k such that A177894(k) = 0.

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Mathematica

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A286138 Pseudo-palindromic numbers: not palindromes (A002113), but a nontrivial palindromic concatenation (AA or ABA) of arbitrary nonzero integers A and B.

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PARI

A369347 Numbers whose decimal expansion is quasiperiodic.

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PARI