A037042 -id:A037042

Search: a037042 -id:a037042

Displaying 1-10 of 16 results found.

page 1 2

A006886

Kaprekar numbers: positive numbers 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.
(Formerly M4625)

+10
26

1, 9, 45, 55, 99, 297, 703, 999, 2223, 2728, 4879, 4950, 5050, 5292, 7272, 7777, 9999, 17344, 22222, 38962, 77778, 82656, 95121, 99999, 142857, 148149, 181819, 187110, 208495, 318682, 329967, 351352, 356643, 390313, 461539, 466830, 499500, 500500, 533170

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

4879 and 5292 are in this sequence but not in A053816.

Digital root is either 1 or 9. - Ezhilarasu Velayutham, Jul 27 2019

Named after the Indian recreational mathematician Dattatreya Ramchandra Kaprekar (1905-1986). - Amiram Eldar, Jun 19 2021

REFERENCES

D. R. Kaprekar, On Kaprekar numbers, J. Rec. Math., Vol. 13 (1980-1981), pp. 81-82.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Penguin Books, NY, 1986, p. 151.

LINKS

Robert Gerbicz, Table of n, a(n) for n = 1..51514 [T. D. Noe computed terms 1 - 1019, Nov 10 2007; R. Gerbicz computed the first 51514 terms, Jul 28 2011]

Santanu Bandyopadhyay, Kaprekar Number, Indian Institute of Technology Bombay (Mumbai, India, 2020).

Hans Havermann, The first 11 million Kaprekar numbers (plus the region around the billionth).

Douglas E. Iannucci, The Kaprekar Numbers, Journal of Integer Sequences, Vol. 3 (2000), Article 1.2,

Douglas E. Iannucci and Bertrum Foster, Kaprekar Triples, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.8.

Robert Munafo, Kaprekar Sequences.

Rosetta Code, Kaprekar numbers.

Walter Schneider, Kaprekar Numbers, 2002.

Gérard Villemin's Almanach of Numbers, Nombres de Kaprekar

Eric Weisstein's World of Mathematics, Kaprekar Number.

Wikipedia, Kaprekar number.

Index entries for Colombian or self numbers and related sequences

FORMULA

a(n) = A194218(n) + A194219(n) and A194218(n) concatenated with A194219(n) gives a(n)^2. - Reinhard Zumkeller, Aug 19 2011

EXAMPLE

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

MATHEMATICA

(* This Mathematica code computes five additional powers in order to be sure that all the Kaprekar numbers have been computed. This fix works for mx <= 50, which includes terms computed by Gerbicz. *)

Inv[a_, b_] := PowerMod[a, -1, b]; mx = 20; t = {1}; Do[h = 10^k - 1; d = Divisors[h]; d2 = Select[d, GCD[#, h/#] == 1 &]; If[Log[10, h] < mx, AppendTo[t, h]]; Do[q = d2[[i]]*Inv[d2[[i]], h/d2[[i]]]; If[Log[10, q] < mx, AppendTo[t, q]], {i, 2, Length[d2] - 1}], {k, mx + 5}]; t = Union[t] (* T. D. Noe, Aug 17 2011, Aug 18 2011 *)

kaprQ[\[Nu]_] := Module[{n = \[Nu]^2},

MemberQ[Plus @@ # & /@

Select[Table[{Floor[n/10^j], 10^j*FractionalPart[n/10^j]}, {j,

IntegerLength@n - 1}], #[[2]] != 0 &], \[Nu]]];

Select[Range@1000000, kaprQ] (* Hans Rudolf Widmer, Oct 22 2021 *)

PROG

(Haskell) -- See A194218 for another version

a006886 n = a006886_list !! (n-1)

a006886_list = 1 : filter chi [4..] where

chi n = read (reverse us) + read (reverse vs) == n where

(us, vs) = splitAt (length $ show n) (reverse $ show (n^2))

-- Reinhard Zumkeller, Aug 18 2011

CROSSREFS

See A053816 for another version.

Cf. A037042, A053394, A053395, A053396, A053397, A045913, A003052.

Cf. A193992 (where 10^n-1 occurs in A006886), A194232 (first differences).

Subsequence of A248353.

KEYWORD

nonn,nice,base,easy

AUTHOR

Robert Munafo

EXTENSIONS

More terms from Michel ten Voorde, Apr 11 2001

4879 and 5292 added by Larry Reeves (larryr(AT)acm.org), Apr 24 2001

38962 added by Larry Reeves (larryr(AT)acm.org), May 23 2002

STATUS

approved

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.

+10
8

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

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

REFERENCES

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.

LINKS

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

EXAMPLE

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.

CROSSREFS

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

KEYWORD

nonn,base,easy

AUTHOR

N. J. A. Sloane

EXTENSIONS

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

STATUS

approved

A053394

The full list of 3-Kaprekar numbers.

+10
7

1, 297, 703, 999

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

Table of n, a(n) for n=1..4.

D. E. Iannucci, The Kaprekar numbers, J. Integer Sequences, Vol. 3, 2000, #1.2.

Index entries for Colombian or self numbers and related sequences

CROSSREFS

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

KEYWORD

nonn,fini,full,base

AUTHOR

N. J. A. Sloane, Jan 07 2000

STATUS

approved

A053395

Complete list of 4-Kaprekar numbers.

+10
7

1, 2223, 2728, 4950, 5050, 7272, 7777, 9999

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

Table of n, a(n) for n=1..8.

D. E. Iannucci, The Kaprekar numbers, J. Integer Sequences, Vol. 3, 2000, #1.2.

Index entries for Colombian or self numbers and related sequences

CROSSREFS

Cf. A006886, A037042, A053394, A053396, A053397, A333133.

KEYWORD

nonn,fini,full,base

AUTHOR

N. J. A. Sloane, Jan 07 2000

STATUS

approved

A053396

The full list of 5-Kaprekar numbers.

+10
7

1, 4879, 17344, 22222, 77778, 82656, 95121, 99999

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

Table of n, a(n) for n=1..8.

D. E. Iannucci, The Kaprekar numbers, J. Integer Sequences, Vol. 3, 2000, #1.2.

Index entries for Colombian or self numbers and related sequences

CROSSREFS

Cf. A006886, A037042, A053394, A053395, A053397, A333133.

KEYWORD

nonn,fini,full,base

AUTHOR

N. J. A. Sloane, Jan 07 2000

STATUS

approved

A053397

The full list of 6-Kaprekar numbers.

+10
7

1, 5292, 38962, 142857, 148149, 181819, 187110, 208495, 318682, 329967, 351352, 356643, 390313, 461539, 466830, 499500, 500500, 533170, 538461, 609687, 643357, 648648, 670033, 681318, 791505, 812890, 818181, 851851, 857143, 961038, 994708, 999999

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

Table of n, a(n) for n=1..32.

D. E. Iannucci, The Kaprekar numbers, J. Integer Sequences, Vol. 3, 2000, #1.2.

Index entries for Colombian or self numbers and related sequences

CROSSREFS

Cf. A006886, A037042, A053394, A053395, A053396, A333133.

KEYWORD

nonn,fini,full,base

AUTHOR

N. J. A. Sloane, Jan 07 2000

STATUS

approved

A274833

6-white numbers: partition digits of n^6 into blocks of 6 starting at right; sum of these 6-digit numbers equals n.

+10
7

0, 1, 1208494, 1358344, 1415583, 1538460, 1734265, 1773226, 1818180, 1994707, 2155140, 2187108, 2208493, 2215486, 2272725, 2272726, 2311687, 2318680, 2351350, 2356641, 2358343, 2363634, 2390311, 2402596, 2420874, 2449252, 2454544, 2459835, 2481220, 2500498, 2533168

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

COMMENTS

Three pairs of consecutive terms: 2272725 and 2272726; 2999997 and 2999998; 3272724 and 3272725.

LINKS

Paolo P. Lava, Table of n, a(n) for n = 1..104

EXAMPLE

1208494^6 = 3115064124992224583219040254156270656 and 3 + 115064 + 124992 + 224583 + 219040 + 254156 + 270656 = 1208494.

MAPLE

P:=proc(q, h) local a, b, n;

for n from 0 to q do a:=n^h; b:=0; while a>0 do b:=b+(a mod 10^h); a:=trunc(a/10^h); od;

if n=b then print(n); fi; od; end: P(10^6, 6);

MATHEMATICA

k = 6; Select[Range[0, 10^7], Function[n, Total[FromDigits /@ Partition[PadLeft[#, Length@ # + k - Mod[Length@ #, k]], k]] == n &@ IntegerDigits[n^k]]] (* Michael De Vlieger, Jul 08 2016, after Harvey P. Dale at A037045 *)

CROSSREFS

Cf. A037042-A037045, A274834.

KEYWORD

nonn,base,fini,full

AUTHOR

Paolo P. Lava, Jul 08 2016

STATUS

approved

A274834

7-white numbers: partition digits of n^7 into blocks of 7 starting at right; sum of these 7-digit numbers equals n.

+10
7

0, 1, 11111110, 13477450, 20483494, 22705717, 24588560, 25411435, 26522546, 27150160, 27150161, 27777775, 28261271, 28744768, 28888885, 28888886, 29372382, 29372383, 29516500, 29855879, 31111109, 31738723, 32078101, 32222218, 32705716, 32849833, 33189212, 33333331

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

COMMENTS

Seven pairs of consecutive numbers: 27150160 and 27150161; 28888885 and 28888886; 29372382 and 29372383; 35555551 and 35555552; 37777774 and 37777775; 40483492 and 40483493; 41111107 and 41111108.

Two sets of three consecutive numbers: 34444440, 34444441 and 34444442; 39999995, 39999996 and 39999997.

LINKS

Paolo P. Lava, Table of n, a(n) for n = 1..63

EXAMPLE

11111110^7 = 20907501177620218737880174500399224623868710000000 and

2 + 0907501 + 1776202 + 1873788 + 0174500 + 3992246 + 2386871 + 0000000 = 11111110.

MAPLE

P:=proc(q, h) local a, b, n;

for n from 0 to q do a:=n^h; b:=0; while a>0 do b:=b+(a mod 10^h); a:=trunc(a/10^h); od;

if n=b then print(n); fi; od; end: P(10^6, 7);

CROSSREFS

Cf. A037042-A037045, A274833.

KEYWORD

nonn,base,fini,full

AUTHOR

Paolo P. Lava, Jul 08 2016

STATUS

approved

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.

+10
6

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

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

REFERENCES

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.

LINKS

Chai Wah Wu, Table of n, a(n) for n = 1..50000

D. 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

EXAMPLE

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

MATHEMATICA

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 *)

PROG

(Haskell)

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

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

CROSSREFS

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

KEYWORD

nonn,nice,base,easy

AUTHOR

Robert Munafo

EXTENSIONS

More terms from Michel ten Voorde, Apr 11 2001

STATUS

approved

A333133

7-Kaprekar numbers.

+10
4

1, 627615, 4444444, 4927941, 5072059, 5555556, 9372385, 9999999

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

No n-Kaprekar number k can have more than n digits because then the number to the left of the plus sign would have more digits than k itself, meaning the sum will always be greater than k.

LINKS

Table of n, a(n) for n=1..8.

EXAMPLE

627615 is in this sequence because inserting a + before the 7th digit from the right of 627615^2 = 393900588225 yields 39390 + 0588225, which equals 627615 (the starting number).

CROSSREFS

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

KEYWORD

nonn,fini,full,base

AUTHOR

Eric Fox, Mar 09 2020

STATUS

approved

page 1 2

Search completed in 0.013 seconds