A006886 -id:A006886 - cp's OEIS frontend

A053816 Another version of the Kaprekar numbers (A006886): n such that n = q+r and n^2 = q*10^m+r, for some m >= 1, q >= 0 and 0 <= r < 10^m, with n != 10^a, a >= 1 and n an m-digit number.

1, 9, 45, 55, 99, 297, 703, 999, 2223, 2728, 4950, 5050, 7272, 7777, 9999, 17344, 22222, 77778, 82656, 95121, 99999, 142857, 148149, 181819, 187110, 208495, 318682, 329967, 351352, 356643, 390313, 461539, 466830, 499500, 500500, 533170, 538461, 609687, 643357

Offset: 1

Robert Munafo

Consider an m-digit number n. Square it and add the right m digits to the left m or m-1 digits. If the resultant sum is n, then n is a term of the sequence.
4879 and 5292 are in A006886 but not in this version.
Shape of plot (see links) seems to consist of line segments whose lengths along the x-axis depend on the number of unitary divisors of 10^m-1 which is equal to 2^w if m is a multiple of 3 or 2^(w+1) otherwise, where w is the number of distinct prime factors of the repunit of length m (A095370). w for m = 60 is 20, whereas w <= 15 for m < 60. This leads to the long segment corresponding to m = 60. - Chai Wah Wu, Jun 02 2016
If n*(n-1) is divisible by 10^m-1 then n is a term where m is the number of decimal digits in n. - Giorgos Kalogeropoulos, Mar 27 2025

			703 is Kaprekar because 703 = 494 + 209, 703^2 = 494209.

D. R. Kaprekar, On Kaprekar numbers, J. Rec. Math., 13 (1980-1981), 81-82.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Penguin Books, NY, 1986, p. 151.

Chai Wah Wu, Table of n, a(n) for n = 1..50000
Shyam Sunder Gupta, On Some Marvellous Numbers of Kaprekar, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 9, 275-315.
Douglas E. Iannucci, The Kaprekar numbers, J. Integer Sequences, Vol. 3, 2000, #1.2.
Robert Munafo, Kaprekar Sequences.
Eric Weisstein's World of Mathematics, Kaprekar Number.
Chai Wah Wu, Semilog plot of A053816(n), n = 1..1003371 (all terms with m <= 60).
Chai Wah Wu, Plot of A053816(n), n = 1..1003371 (all terms with m <= 60).
Index entries for Colombian or self numbers and related sequences

Cf. A006886, A037042, A053394, A053395, A053396, A053397, A045913, A003052, A055642, A095370.

Fixed points of A380585.

a053816 n = a053816_list !! (n-1)
a053816_list = 1 : filter f [4..] where
   f x = length us - length vs <= 1 &&
         read (reverse us) + read (reverse vs) == x
         where (us, vs) = splitAt (length $ show x) (reverse $ show (x^2))
-- Reinhard Zumkeller, Oct 04 2014

kapQ[n_]:=Module[{idn2=IntegerDigits[n^2],len},len=Length[idn2];FromDigits[ Take[idn2,Floor[len/2]]]+FromDigits[Take[idn2, -Ceiling[len/2]]]==n]; Select[Range[540000],kapQ] (* Harvey P. Dale, Aug 22 2011 *)
ktQ[n_] := ((x = n^2) - (z = FromDigits[Take[IntegerDigits[x], y = -IntegerLength[n]]]))*10^y + z == n; Select[Range[540000], ktQ] (* Jayanta Basu, Aug 04 2013 *)
Select[Range[540000],Total[FromDigits/@TakeDrop[IntegerDigits[#^2], Floor[ IntegerLength[ #^2]/2]]] ==#&] (* The program uses the TakeDrop function from Mathematica version 10 *) (* Harvey P. Dale, Jun 03 2016 *)
maxDigits=6; Flatten[Table[lst={};sub=Subsets@FactorInteger[v=10^d-1]; Do[a=Times@@Power@@@s; n=ChineseRemainder[{0,1},{a,v/a},1]; If[10^(d-1)<=n<10^d,AppendTo[lst,n]],{s,sub}];Union@lst,{d,maxDigits}]] (* Giorgos Kalogeropoulos, Mar 27 2025 *)

isok(n) = n == vecsum(divrem(n^2, 10^(1+logint(n, 10)))); \\ Ruud H.G. van Tol, Jun 02 2024

def is_A053816(n): return n==sum(divmod(n**2,10**len(str(n)))) and n
print(upto_1e5:=list(filter(is_A053816, range(10**5)))) # M. F. Hasler, Mar 28 2025

More terms from Michel ten Voorde, Apr 11 2001

A193992 Position where 10^n-1 occurs in the Kaprekar sequence A006886.

Original entry on oeis.org

2, 5, 8, 17, 24, 54, 62, 91, 102, 132, 149, 264, 281, 316, 385, 503, 527, 762, 790, 1035, 1154, 1278, 1378, 2304, 2374, 2498, 2575, 3122, 3910, 11330, 11714, 15400, 15478, 15642, 16039, 17892, 17909, 17968, 18401, 22238, 23747, 38524, 38728, 40625, 41101

Offset: 1

T. D. Noe, Aug 17 2011

Partial sums of A194232.

The Mathematica code computes 50 terms, but only these 45 terms are correct.

Hans Havermann, Table of n, a(n) for n = 1..196
Index entries for Colombian or self numbers and related sequences

Cf. A006886, A194232.

Inv[a_, b_] := PowerMod[a, -1, b]; t = {1}; Do[h = 10^k-1; d = Divisors[h]; d2 = Select[d, GCD[#, h/#] == 1 &]; AppendTo[t, h]; Do[AppendTo[t, d2[[i]]*Inv[d2[[i]], h/d2[[i]]]], {i, 2, Length[d2]-1}], {k, 50}]; t = Union[t]; Table[Position[t, 10^n-1, 1, 1][[1,1]], {n, Log[10, t[[-1]]]}]

A194232 Number of n-digit Kaprekar numbers (A006886).

Original entry on oeis.org

2, 3, 3, 9, 7, 30, 8, 29, 11, 30, 17, 115, 17, 35, 69, 118, 24, 235, 28, 245, 119, 124, 100, 926, 70, 124, 77, 547, 788, 7420, 384, 3686, 78, 164, 397, 1853, 17, 59, 433, 3837, 1509, 14777, 204, 1897, 476, 185, 748, 7390, 213, 1877, 320, 963, 421, 3812, 1190

Offset: 1

Hans Havermann, Aug 19 2011

First difference of A193992. - Hans Havermann, Aug 19 2011

Hans Havermann, Table of n, a(n) for n = 1..196
Index entries for Colombian or self numbers and related sequences

Cf. A006886, A193992.

A382160 Kaprekar numbers according to the definition in A006886 that are not in A053816.

Original entry on oeis.org

4879, 5292, 38962, 627615, 5479453, 8161912, 243902440, 665188470, 867208672, 909090909, 2646002646, 7359343993, 8975672343, 19481019481, 65098401732, 71428071429, 74074074075, 74761738129, 81433418067, 81933418567, 90909090910, 93555093555, 98268434902, 218400870420

Offset: 1

N. J. A. Sloane, Mar 25 2025

Amiram Eldar, Table of n, a(n) for n = 1..5151
Shyam Sunder Gupta, On Some Marvellous Numbers of Kaprekar, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 9, 275-315.

Cf. A006886, A053816.

More terms from Amiram Eldar, Mar 26 2025

A382165 Kaprekar numbers (A006886) that are divisible by the sum of their digits.

Original entry on oeis.org

1, 9, 45, 999, 2223, 4950, 5050, 5292, 7272, 142857, 148149, 187110, 356643, 466830, 499500, 500500, 538461, 627615, 648648, 681318, 791505, 818181, 961038, 994708, 5555556, 11111112, 16590564, 30884184, 36363636, 49995000, 50005000, 55474452, 74747475, 234567901, 432432432, 665188470, 999999999, 2020202020, 3846956652, 4132841328, 4999950000, 5000050000

Offset: 1

N. J. A. Sloane, Mar 26 2025

Amiram Eldar, Table of n, a(n) for n = 1..5091
Shyam Sunder Gupta, On Some Marvellous Numbers of Kaprekar, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 9, 275-315.

Intersection of A005349 and A006886.

A164704 A006886 mod 9.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 1, 0, 1, 0, 0, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 1, 1

Offset: 1

Paul Curtz, Aug 23 2009

The definition implies that if m is A006886(n) for some n, then m == m^2 mod 9, hence m == 0 or 1 mod 9, as conjectured by Paul Curtz. - N. J. A. Sloane, Aug 23 2009

Edited and extended by N. J. A. Sloane, Aug 23 2009

A003052 Self numbers or Colombian numbers (numbers that are not of the form m + sum of digits of m for any m).

Original entry on oeis.org

1, 3, 5, 7, 9, 20, 31, 42, 53, 64, 75, 86, 97, 108, 110, 121, 132, 143, 154, 165, 176, 187, 198, 209, 211, 222, 233, 244, 255, 266, 277, 288, 299, 310, 312, 323, 334, 345, 356, 367, 378, 389, 400, 411, 413, 424, 435, 446, 457, 468, 479, 490, 501, 512, 514, 525

Offset: 1

N. J. A. Sloane

From Amiram Eldar, Nov 28 2020: (Start)
The term "self numbers" was coined by Kaprekar (1959). The term "Colombian number" was coined by Recamán (1973) of Bogota, Colombia.
The asymptotic density of this sequence is approximately 0.0977778 (Guaraldo, 1978). (End)

Steven R. Finch, Mathematical Constants, Cambridge, 2003, Section 2.24.
Martin Gardner, Time Travel and Other Mathematical Bewilderments. Freeman, NY, 1988, p. 116.
V. S. Joshi, A note on self-numbers. Volume dedicated to the memory of V. Ramaswami Aiyar. Math. Student, Vol. 39 (1971), pp. 327-328. MR0330032 (48 #8371).
D. R. Kaprekar, Puzzles of the Self-Numbers. 311 Devlali Camp, Devlali, India, 1959.
D. R. Kaprekar, The Mathematics of the New Self Numbers, Privately Printed, 311 Devlali Camp, Devlali, India, 1963.
D. R. Kaprekar, The Mathematics of the New Self Numbers (Part V). 311 Devlali Camp, Devlali, India, 1967.
Bernardo Recamán, The Bogota Puzzles, Dover Publications, Inc., 2020, chapter 36, p. 33.
József Sándor and Borislav Crstici, Handbook of Number theory II, Kluwer Academic Publishers, 2004, Chapter 4, pp. 384-386.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Author?, J. Recreational Math., vol. 23, no. 1, p. 244, 1991.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Max A. Alekseyev and N. J. A. Sloane, On Kaprekar's Junction Numbers, arXiv:2112.14365, 2021; Journal of Combinatorics and Number Theory 12:3 (2022), 115-155.
Christian N. K. Anderson, Ulam Spiral of the first 5000 self numbers.
Santanu Bandyopadhyay, Self-Number, Indian Institute of Technology Bombay (Mumbai, India, 2020).
Santanu Bandyopadhyay, Self-Number, Indian Institute of Technology Bombay (Mumbai, India, 2020). [Local copy]
Martin Gardner and N. J. A. Sloane, Correspondence, 1973-74
Rosalind Guaraldo, On the Density of the Image Sets of Certain Arithmetic Functions - II, The Fibonacci Quarterly, Vol. 16, No. 5 (1978), pp. 481-488.
Shyam Sunder Gupta, On Some Marvellous Numbers of Kaprekar, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 9, 275-315.
D. R. Kaprekar, The Mathematics of the New Self Numbers, 1963. [annotated and scanned]
Andrzej Makowski, On Kaprekar's "junction numbers", Math. Student, Vol. 34, No. 2 (1966), p. 77. MR0223292 (36 #6340); entire issue.
A. Narasinga Rao, On a technique for obtaining numbers with a multiplicity of generators, Math. Student, Vol. 34, No. 2 (1966), pp. 79-84. MR0229573 (37 #5147); entire issue.
Bernardo Recamán, Problem E2408, Amer. Math. Monthly, Vol. 80, No. 4 (1973), p. 434; Colombian Numbers, solution to Problem E2408 by D. W. Bange, ibid., Vol. 81, No. 4 (1974), p. 407.
Giovanni Resta, Self or Colombian numbers, Numbersaplenty, 2013.
Richard Schorn, Kaprekar's Sequence and his "Selfnumbers", DERIVE Newsletter, #53 (2004), pp. 30-32.
Walter Schneider, Self Numbers, 2000-2003.
Walter Schneider, Self Numbers, 2000-2003 (unpublished; local copy)
N. J. A. Sloane, Martin Gardner and D. R. Kaprekar, Correspondence, 1974 [Scanned letters]
Terry Trotter, Charlene Numbers
Eric Weisstein's World of Mathematics, Self Number.
Wikipedia, Self number.
U. Zannier, On the distribution of self-numbers, Proc. Amer. Math. Soc., Vol. 85, No. 1 (1982), pp. 10-14.
Index entries for Colombian or self numbers and related sequences

Cf. A006886, A232229, A062028, A055642, A282711.
For self primes, i.e., self numbers which are primes, see A006378.
Complement of A176995.
See A010061 for the binary version, A283002 for a base-100 version.
Cf. A247104 (subsequence of squarefree terms).
Cf. A377472 for first differences, A377474 for indices where new differences appear.

a003052 n = a003052_list !! (n-1)
a003052_list = filter ((== 0) . a230093) [1..]
-- Reinhard Zumkeller, Oct 11 2013, Aug 21 2011

isA003052 := proc(n) local k ; for k from 0 to n do if k+A007953(k) = n then RETURN(false): fi; od: RETURN(true) ; end:
A003052 := proc(n) option remember; if n = 1 then 1; else for a from procname(n-1)+1 do if isA003052(a) then RETURN(a) ; fi; od; fi; end: # R. J. Mathar, Jul 27 2009

nn = 525; Complement[Range[nn], Union[Table[n + Total[IntegerDigits[n]], {n, nn}]]] (* T. D. Noe, Mar 31 2013 *)

is_A003052(n)={for(i=1,min(n\2,9*#digits(n)), sumdigits(n-i)==i && return); n}  \\ M. F. Hasler, Mar 20 2011, updated Nov 08 2018

is(n) = {if(n < 30, return((n < 10 && n%2 == 1) || n == 20)); qd = 1 + logint(n, 10); r = 1 + (n-1)%9; h = (r + 9 * (r%2))/2; ld = 10; while(h + 9*qd >= n % ld, ld*=10); vs = vecsum(digits(n \ ld)); n %= ld; for(i = 0, qd, if(vs + vecsum(digits(n - h - 9*i)) == h + 9*i, return(0))); 1} \\ David A. Corneth, Aug 20 2020

A230093(a(n)) = 0. - Reinhard Zumkeller, Oct 11 2013

In fact this defines the sequence: x is in the sequence iff A230093(x) = 0. - M. F. Hasler, Nov 08 2018

More terms from James Sellers, Jul 06 2000

A045913 Kaprekar numbers: numbers k such that k = q + r and k^2 = q*10^m + r, for some m >= 1, q >= 0 and 0 <= r < 10^m. Here q and r must both have the same number of digits.

Original entry on oeis.org

1, 9, 45, 55, 703, 4950, 5050, 7272, 7777, 77778, 82656, 318682, 329967, 351352, 356643, 390313, 461539, 466830, 499500, 500500, 533170, 538461, 609687, 643357, 648648, 670033, 681318, 791505, 812890, 818181, 851851, 857143, 4444444, 4927941, 5072059, 5555556, 11111112, 36363636, 38883889, 44363341, 44525548, 49995000, 50005000

Offset: 1

N. J. A. Sloane

A variant of Kaprekar's original definition (A006886).

			703 is Kaprekar because 703 = 494 + 209, 703^2 = 494209.
11111112^2 = 123456809876544 = (1234568 + 9876544)^2. The two "halves" of the square have the same length here, although it's not m but rather m - 1.

D. R. Kaprekar, On Kaprekar numbers, J. Rec. Math., 13 (1980-1981), 81-82.
D. Wells, The Penguin Dictionary of Curious and Interesting Numbers, Penguin Books, NY, 1986, p. 151.

Jinyuan Wang, Table of n, a(n) for n = 1..30047
D. E. Iannucci, The Kaprekar numbers, J. Integer Sequences, Vol. 3, 2000, #1.2.
Rosetta Code, Kaprekar numbers
Eric Weisstein's World of Mathematics, Kaprekar Number
Wikipedia, Kaprekar number
Index entries for Colombian or self numbers and related sequences

Cf. A006886, A037042, A053394, A053395, A053396, A053397, A003052, A248353.

More terms from Michel ten Voorde, Apr 13 2001
Definition clarified by Reinhard Zumkeller, Oct 05 2014
Definition modified and terms corrected by Max Alekseyev, Aug 06 2017

A238237 Numbers which when chopped into two parts with equal length, added and squared result in the same number.

Original entry on oeis.org

81, 2025, 3025, 9801, 494209, 998001, 24502500, 25502500, 52881984, 60481729, 99980001, 6049417284, 6832014336, 9048004641, 9999800001, 101558217124, 108878221089, 123448227904, 127194229449, 152344237969, 213018248521, 217930248900, 249500250000, 250500250000

Offset: 1

Arkadiusz Wesolowski, Feb 20 2014

Yet another variant of the Kaprekar numbers A006886. - N. J. A. Sloane, Aug 06 2017
From Bernard Schott, Jan 21 2022: (Start)
Three subsequences:
-> {(10^m-1)^2, m >= 1} = A059988 \ {0}; see example 9801.
-> {(10^m-1)^2 * 10^(2*m) / 4, m >= 1} = A350869 \ {0}; see example 2025.
-> {(10^m+1)^2 * 10^(2*m) / 4, m >= 1} = A038544 \ {1}, see example 3025. (End)

			2025 = (20 + 25)^2, so 2025 is in the sequence.
3025 = (30 + 25)^2, so 3025 is in the sequence.
9801 = (98 + 01)^2, so 9801 is in the sequence.

Rémy Sigrist, Table of n, a(n) for n = 1..25000

Subsequence of A102766.
Subsequence: A350870.
Cf. A006886, A038544, A059988, A350869.
For square roots see A290449.

Select[Range[600000]^2, EvenQ[len=IntegerLength[#]] && # == (Mod[#,10^(len/2)] + Floor[#/10^(len/2)])^2 &] (* Stefano Spezia, Jan 01 2025 *)

forstep(m=1, 7, 2, p=10^((m+1)/2); for(n=10^m, 10^(m+1)-1, d=lift(Mod(n, p)); if(((n-d)/p+d)^2==n, print1(n, ", "))));

a(n) = A290449(n)^2. - Bernard Schott, Jan 20 2022

a(12)-a(24) from Donovan Johnson, Feb 22 2014

A053394 The full list of 3-Kaprekar numbers.

Original entry on oeis.org

1, 297, 703, 999

Offset: 1

N. J. A. Sloane, Jan 07 2000

D. E. Iannucci, The Kaprekar numbers, J. Integer Sequences, Vol. 3, 2000, #1.2.
Index entries for Colombian or self numbers and related sequences

Cf. A006886, A037042, A053395, A053396, A053397, A333133.

cp's OEIS Frontend

A053816 Another version of the Kaprekar numbers (A006886): n such that n = q+r and n^2 = q*10^m+r, for some m >= 1, q >= 0 and 0 <= r < 10^m, with n != 10^a, a >= 1 and n an m-digit number.

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A193992 Position where 10^n-1 occurs in the Kaprekar sequence A006886.

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A194232 Number of n-digit Kaprekar numbers (A006886).

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A382160 Kaprekar numbers according to the definition in A006886 that are not in A053816.

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A382165 Kaprekar numbers (A006886) that are divisible by the sum of their digits.

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A164704 A006886 mod 9.

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A003052 Self numbers or Colombian numbers (numbers that are not of the form m + sum of digits of m for any m).

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A045913 Kaprekar numbers: numbers k such that k = q + r and k^2 = q*10^m + r, for some m >= 1, q >= 0 and 0 <= r < 10^m. Here q and r must both have the same number of digits.

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A238237 Numbers which when chopped into two parts with equal length, added and squared result in the same number.

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