A020338 -id:A020338 - cp's OEIS frontend

A381917 Kaprekar numbers that are the concatenation of two equal numbers.

55, 99, 5050, 7272, 7777, 9999, 500500, 648648, 851851, 999999, 13641364, 24752475, 25252525, 36363636, 50005000, 61116111, 88888888, 99999999, 1111111111, 3888938889, 4132841328, 5000050000, 5243952439, 9756097560, 9999999999, 159341159341, 175676175676, 233415233415

Offset: 1

Shyam Sunder Gupta, Mar 10 2025

All repdigit numbers of even length which are Kaprekar numbers are terms. Since 99, 9999, 999999, ... and 5050, 500500, 50005000, ... are Kaprekar numbers, there are infinitely many terms.

			648648 is a Kaprekar number which is the concatenation of 648 and 648.

Shyam Sunder Gupta, Table of n, a(n) for n = 1..2163
Shyam Sunder Gupta, On Some Marvellous Numbers of Kaprekar, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 9, 275-315.

Intersection of A006886 and A020338.

Cf. A092118.

A249166 Odd integers concatenated with themselves.

Original entry on oeis.org

11, 33, 55, 77, 99, 1111, 1313, 1515, 1717, 1919, 2121, 2323, 2525, 2727, 2929, 3131, 3333, 3535, 3737, 3939, 4141, 4343, 4545, 4747, 4949, 5151, 5353, 5555, 5757, 5959, 6161, 6363, 6565, 6767, 6969, 7171, 7373, 7575, 7777, 7979, 8181, 8383, 8585, 8787

Offset: 1

M. F. Hasler, Oct 22 2014

Bisection (every other term) of A020338. See A248422 for the complement.

Cf. A020338, A248422.

[Seqint(Intseq(n) cat Intseq(n)): n in [1..99 by 2]]

PARI
```
a(n) = eval(Str(2*n-1,2*n-1))
```

A275238 a(n) = n*(10^floor(log_10(n)+1) + 1) + (-1)^n.

Original entry on oeis.org

1, 10, 23, 32, 45, 54, 67, 76, 89, 98, 1011, 1110, 1213, 1312, 1415, 1514, 1617, 1716, 1819, 1918, 2021, 2120, 2223, 2322, 2425, 2524, 2627, 2726, 2829, 2928, 3031, 3130, 3233, 3332, 3435, 3534, 3637, 3736, 3839, 3938, 4041, 4140, 4243, 4342, 4445, 4544, 4647, 4746, 4849, 4948, 5051, 5150, 5253, 5352, 5455, 5554

Offset: 0

Ilya Gutkovskiy, Jul 21 2016

Concatenation of n with n+(-1)^n (A004442).
Subsequence of A248378.
Primes in this sequence: 23, 67, 89, 1213, 3637, 4243, 5051, 5657, 6263, 6869, 7879, 8081, 9091, 9293, 9697, 102103, ... (A030458).
Numbers n such that a(n) is prime: 2, 6, 8, 12, 36, 42, 50, 56, 62, 68, 78, 80, 90, 92, 96, 102, 108, 120, 126, 138, ... (A030457).

			a(0) =  0 + 1 = 1;
a(1) = 11 - 1 = 10;
a(2) = 22 + 1 = 23;
a(3) = 33 - 1 = 32;
a(4) = 44 + 1 = 45;
a(5) = 55 - 1 = 54, etc.
or
a(0) =  1 -> concatenation of 0 with 0 + (-1)^0 = 1;
a(1) = 10 -> concatenation of 1 with 1 + (-1)^1 = 0;
a(2) = 23 -> concatenation of 2 with 2 + (-1)^2 = 3;
a(3) = 32 -> concatenation of 3 with 3 + (-1)^3 = 2;
a(4) = 45 -> concatenation of 4 with 4 + (-1)^4 = 5;
a(5) = 54 -> concatenation of 5 with 5 + (-1)^5 = 4, etc.
........................................................
a(2k) = 1, 23, 45, 67, 89, 1011, 1213, 1415, 1617, 1819, ...

Cf. A001704, A002285, A004442, A020338, A030457, A030458, A030656, A033999, A064834, A248378.

Table[n (10^Floor[Log[10, n] + 1] + 1) + (-1)^n, {n, 0, 55}]

a(n) = if(n, n*(10^(logint(n,10)+1) + 1) + (-1)^n, 1) \\ Charles R Greathouse IV, Jul 21 2016

a(n) = A020338(n) + A033999(n).
a(2k) = A030656(k).
A064834(a(n)) > 0, for n > 0.
a(n) ~ 10*n*10^floor(c*log(n)), where c = 1/log(10) = 0.4342944819... = A002285.

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)) }

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A381917 Kaprekar numbers that are the concatenation of two equal numbers.

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A249166 Odd integers concatenated with themselves.

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A275238 a(n) = n*(10^floor(log_10(n)+1) + 1) + (-1)^n.

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Mathematica

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A369347 Numbers whose decimal expansion is quasiperiodic.

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