cp's OEIS Frontend

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

%I A003052 M2404 #172 Jul 02 2025 16:01:54
%S A003052 1,3,5,7,9,20,31,42,53,64,75,86,97,108,110,121,132,143,154,165,176,
%T A003052 187,198,209,211,222,233,244,255,266,277,288,299,310,312,323,334,345,
%U A003052 356,367,378,389,400,411,413,424,435,446,457,468,479,490,501,512,514,525
%N A003052 Self numbers or Colombian numbers (numbers that are not of the form m + sum of digits of m for any m).
%C A003052 From _Amiram Eldar_, Nov 28 2020: (Start)
%C A003052 The term "self numbers" was coined by Kaprekar (1959). The term "Colombian number" was coined by Recamán (1973) of Bogota, Colombia.
%C A003052 The asymptotic density of this sequence is approximately 0.0977778 (Guaraldo, 1978). (End)
%D A003052 Steven R. Finch, Mathematical Constants, Cambridge, 2003, Section 2.24.
%D A003052 Martin Gardner, Time Travel and Other Mathematical Bewilderments. Freeman, NY, 1988, p. 116.
%D A003052 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 A003052 D. R. Kaprekar, Puzzles of the Self-Numbers. 311 Devlali Camp, Devlali, India, 1959.
%D A003052 D. R. Kaprekar, The Mathematics of the New Self Numbers, Privately Printed, 311 Devlali Camp, Devlali, India, 1963.
%D A003052 D. R. Kaprekar, The Mathematics of the New Self Numbers (Part V). 311 Devlali Camp, Devlali, India, 1967.
%D A003052 Bernardo Recamán, The Bogota Puzzles, Dover Publications, Inc., 2020, chapter 36, p. 33.
%D A003052 József Sándor and Borislav Crstici, Handbook of Number theory II, Kluwer Academic Publishers, 2004, Chapter 4, pp. 384-386.
%D A003052 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D A003052 Author?, J. Recreational Math., vol. 23, no. 1, p. 244, 1991.
%H A003052 Reinhard Zumkeller, <a href="/A003052/b003052.txt">Table of n, a(n) for n = 1..10000</a>
%H A003052 Max A. Alekseyev and N. J. A. Sloane, <a href="https://arxiv.org/abs/2112.14365">On Kaprekar's Junction Numbers</a>, arXiv:2112.14365, 2021; Journal of Combinatorics and Number Theory 12:3 (2022), 115-155.
%H A003052 Christian N. K. Anderson, <a href="/A003052/a003052.jpg">Ulam Spiral</a> of the first 5000 self numbers.
%H A003052 Santanu Bandyopadhyay, <a href="https://www.ese.iitb.ac.in/~santanu/RM8.pdf">Self-Number</a>, Indian Institute of Technology Bombay (Mumbai, India, 2020).
%H A003052 Santanu Bandyopadhyay, <a href="/A003052/a003052_3.pdf">Self-Number</a>, Indian Institute of Technology Bombay (Mumbai, India, 2020). [Local copy]
%H A003052 Martin Gardner and N. J. A. Sloane, <a href="/A003154/a003154.pdf">Correspondence, 1973-74</a>
%H A003052 Rosalind Guaraldo, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/16-5/guaraldo.pdf">On the Density of the Image Sets of Certain Arithmetic Functions - II</a>, The Fibonacci Quarterly, Vol. 16, No. 5 (1978), pp. 481-488.
%H A003052 Shyam Sunder Gupta, <a href="https://doi.org/10.1007/978-981-97-2465-9_9">On Some Marvellous Numbers of Kaprekar</a>, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 9, 275-315.
%H A003052 D. R. Kaprekar, <a href="/A003052/a003052_2.pdf">The Mathematics of the New Self Numbers</a>, 1963. [annotated and scanned]
%H A003052 Andrzej Makowski, On Kaprekar's "junction numbers", Math. Student, Vol. 34, No. 2 (1966), p. 77. MR0223292 (36 #6340); <a href="https://schoolbooksarchive.azimpremjiuniversity.edu.in/handle/20.500.12497/11704">entire issue</a>.
%H A003052 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); <a href="https://schoolbooksarchive.azimpremjiuniversity.edu.in/handle/20.500.12497/11704">entire issue</a>.
%H A003052 Bernardo Recamán, <a href="http://www.jstor.org/stable/2319095">Problem E2408</a>, Amer. Math. Monthly, Vol. 80, No. 4 (1973), p. 434; <a href="http://www.jstor.org/stable/2319017">Colombian Numbers</a>, solution to Problem E2408 by D. W. Bange, ibid., Vol. 81, No. 4 (1974), p. 407.
%H A003052 Giovanni Resta, <a href="http://www.numbersaplenty.com/set/self_number/">Self or Colombian numbers</a>, Numbersaplenty, 2013.
%H A003052 Richard Schorn, <a href="http://www.austromath.at/dug/dnl53.pdf">Kaprekar's Sequence and his "Selfnumbers"</a>, DERIVE Newsletter, #53 (2004), pp. 30-32.
%H A003052 Walter Schneider, <a href="http://web.archive.org/web/2004/www.wschnei.de/digit-related-numbers/self-numbers.html">Self Numbers</a>, 2000-2003.
%H A003052 Walter Schneider, <a href="/A003052/a003052_5.pdf">Self Numbers</a>, 2000-2003 (unpublished; local copy)
%H A003052 N. J. A. Sloane, Martin Gardner and D. R. Kaprekar, <a href="/A003052/a003052_1.pdf">Correspondence, 1974</a> [Scanned letters]
%H A003052 Terry Trotter, <a href="http://www.trottermath.net/numthry/charlene.html">Charlene Numbers</a>
%H A003052 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SelfNumber.html">Self Number</a>.
%H A003052 Wikipedia, <a href="http://en.wikipedia.org/wiki/Self_number">Self number</a>.
%H A003052 U. Zannier, <a href="https://doi.org/10.1090/S0002-9939-1982-0647887-4">On the distribution of self-numbers</a>, Proc. Amer. Math. Soc., Vol. 85, No. 1 (1982), pp. 10-14.
%H A003052 <a href="/index/Coi#Colombian">Index entries for Colombian or self numbers and related sequences</a>
%F A003052 A230093(a(n)) = 0. - _Reinhard Zumkeller_, Oct 11 2013
%F A003052 In fact this defines the sequence: x is in the sequence iff A230093(x) = 0. - _M. F. Hasler_, Nov 08 2018
%p A003052 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:
%p A003052 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
%t A003052 nn = 525; Complement[Range[nn], Union[Table[n + Total[IntegerDigits[n]], {n, nn}]]] (* _T. D. Noe_, Mar 31 2013 *)
%o A003052 (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
%o A003052 (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
%o A003052 (Haskell)
%o A003052 a003052 n = a003052_list !! (n-1)
%o A003052 a003052_list = filter ((== 0) . a230093) [1..]
%o A003052 -- _Reinhard Zumkeller_, Oct 11 2013, Aug 21 2011
%Y A003052 Cf. A006886, A232229, A062028, A055642, A282711.
%Y A003052 For self primes, i.e., self numbers which are primes, see A006378.
%Y A003052 Complement of A176995.
%Y A003052 See A010061 for the binary version, A283002 for a base-100 version.
%Y A003052 Cf. A247104 (subsequence of squarefree terms).
%Y A003052 Cf. A377472 for first differences, A377474 for indices where new differences appear.
%K A003052 nonn,base
%O A003052 1,2
%A A003052 _N. J. A. Sloane_
%E A003052 More terms from _James Sellers_, Jul 06 2000