A003052 - OEIS (original) (raw)

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.

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]

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.

Walter Schneider, Self Numbers, 2000-2003 (unpublished; local copy)

N. J. A. Sloane, Martin Gardner and D. R. Kaprekar, Correspondence, 1974 [Scanned letters]

Eric Weisstein's World of Mathematics, Self Number.

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.

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.