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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.
+20
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
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.
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))
(PARI) isok(n) = n == vecsum(divrem(n^2, 10^(1+logint(n, 10)))); \\ Ruud H.G. van Tol, Jun 02 2024
Position where 10^n-1 occurs in the Kaprekar sequence A006886.
+20
3
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
COMMENTS
The Mathematica code computes 50 terms, but only these 45 terms are correct.
MATHEMATICA
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]]]}]
Number of n-digit Kaprekar numbers ( A006886).
+20
3
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
COMMENTS
First difference of A193992. - Hans Havermann, Aug 19 2011
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
COMMENTS
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
Self numbers or Colombian numbers (numbers that are not of the form m + sum of digits of m for any m).
(Formerly M2404)
+10
90
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
COMMENTS
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)
REFERENCES
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.
Andrzej Makowski, On Kaprekar's "junction numbers", Math. Student, Vol. 34 (1966), p. 77. MR0223292 (36 #6340).
A. Narasinga Rao, On a technique for obtaining numbers with a multiplicity of generators, Math. Student, Vol. 34 (1966), pp. 79-84. MR0229573 (37 #5147).
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.
LINKS
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] 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. Walter Schneider, Self Numbers, 2000-2003 (unpublished; local copy) Terry Trotter, Charlene Numbers [Warning: As of March 2018 this site appears to have been hacked. Proceed with great caution. The original content should be retrieved from the Wayback machine and added here. - N. J. A. Sloane, Mar 29 2018]
FORMULA
In fact this defines the sequence: x is in the sequence iff A230093(x) = 0. - M. F. Hasler, Nov 08 2018
MAPLE
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
MATHEMATICA
nn = 525; Complement[Range[nn], Union[Table[n + Total[IntegerDigits[n]], {n, nn}]]] (* T. D. Noe, Mar 31 2013 *)
PROG
(PARI) 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 (PARI) 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 (Haskell)
a003052 n = a003052_list !! (n-1)
a003052_list = filter ((== 0) . a230093) [1..]
CROSSREFS
For self primes, i.e., self numbers which are primes, see A006378. Cf. A247104 (subsequence of squarefree terms). Cf. A377472 for first differences, A377474 for indices where new differences appear.
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
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.
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.
EXTENSIONS
Definition modified and terms corrected by Max Alekseyev, Aug 06 2017
Numbers which when chopped into two parts with equal length, added and squared result in the same number.
+10
8
81, 2025, 3025, 9801, 494209, 998001, 24502500, 25502500, 52881984, 60481729, 99980001, 6049417284, 6832014336, 9048004641, 9999800001, 101558217124, 108878221089, 123448227904, 127194229449, 152344237969, 213018248521, 217930248900, 249500250000, 250500250000
COMMENTS
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)
EXAMPLE
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.
PROG
(PARI) 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, ", "))));
The full list of 3-Kaprekar numbers.
+10
7
Complete list of 4-Kaprekar numbers.
+10
7
1, 2223, 2728, 4950, 5050, 7272, 7777, 9999
The full list of 5-Kaprekar numbers.
+10
7
1, 4879, 17344, 22222, 77778, 82656, 95121, 99999
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