A003052 Self numbers or Colombian numbers (numbers that are not of the form m + sum of digits of m for any m). (Formerly M2404)

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

1,2

Comments

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)

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

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.

D. R. Kaprekar, The Mathematics of the New Self Numbers, 1963. [annotated and scanned]

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 [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]

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

Formula

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

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..]
-- Reinhard Zumkeller, Oct 11 2013, Aug 21 2011

Crossrefs

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.

Sequence in context: A369612 A083107 A338637 * A003219 A030142 A179085

Adjacent sequences: A003049 A003050 A003051 * A003053 A003054 A003055

Keyword

nonn,base

Author

N. J. A. Sloane

Extensions

More terms from James A. Sellers, Jul 06 2000

Status

approved