A377472 -id:A377472 - cp's OEIS frontend

A377473 Distinct first differences of Colombian or self numbers (A377472), listed in the order they appear.

2, 11, 15, 28, 41, 54, 67, 80, 93, 106, 119, 101, 118, 131, 144, 157, 170, 183, 196, 209, 24, 90, 204, 221, 234, 247, 260, 273, 286, 299, 35, 79, 294, 307, 324, 337, 350, 363, 376, 389, 46, 68, 384, 397, 410, 427, 440, 453, 466, 479, 57, 474, 487, 500

Offset: 1

M. F. Hasler, Oct 30 2024

See A377474 for the indices where these first differences appear for the first time.

			A377472(n) = 2 = a(1) for all n <= 4. Then, A377472(n) = 11 = a(2) up to n = 13.
Then again, A377472(14..23) = (2, 11, ..., 11) and similarly up to n = 94.
But A377472(103) = 15 = a(3). Then the previous pattern repeats, with A377472(n) = 2 for n = 112, 122, ..., 192, followed by A377472(n) = 15 at n = 201, 299, 397, ..., 887.
Then A377472(984) = 28 = a(4), and it goes on with A377472(n) = 2 at n = 992, 1002, ..., 1072, and so on, with A377472(n) = 28 at n = 1962, 2940, 3918, ..., 8808.
Then A377472(9785) = 41 = a(5), and the whole previous pattern repeats, with A377472(9881) = 15, then A377472(10762) = 28 etc.
At n = 97786, we find A377472(n) = 54 = a(6), and again the whole previous pattern repeats again 8 more times, each time separated by a 54, until we have, at n = 977787, A377472(n) = 67 = a(7). And so on.

Daniel Mondot, Table of n, a(n) for n = 1..10000

Cf. A003052 (Colombian numbers), A377472 (1st differences of Colombian numbers), A163139 (= A377472 - 1), A377423.

A377473_upto(N=9, show=1)={my(o, c, d, L=List()); for(n=1+o=1, oo, is_A003052(n)||next; c++; if(!setsearch(L, d=n-o), show && printf("%d, ",[c,d]); listput(L,d); #L

a(n) = A377423(n) + 1.

Terms a(9) onward computed from A377423 by Max Alekseyev, Dec 31 2024

A377474 Indices where new terms arise among first differences of Colombian or self numbers (A377472).

Original entry on oeis.org

1, 5, 103, 984, 9785, 97786, 977787, 9777788, 97777789

Offset: 1

M. F. Hasler, Oct 30 2024

See A377473 for the distinct values of the first differences in the order they appear for the first time.

			The first value, A377472(1) = 2, appears obviously at index a(1) = 1.
The next three values are the same, but at index a(2) = 5 we have a new, distinct value A377472(5) = 11 = A377423(2).
The next distinct value is A377472(103) = 15 = A377423(3), so a(3) = 103.
Then the next new value is A377472(984) = 28 = A377423(4), so a(4) = 984.
The next new value is A377472(9785) = 41 = A377423(5), so a(5) = 9785.
Then, at n = 97786 = a(6), we have A377472(n) = 54 = A377423(6).
Only at n = 977787 = a(7), we have a new value, A377472(n) = 67 = A377423(7).
At n = 9777788 = a(8), we have the next new value, A377472(n) = 80 = A377423(8).

Cf. A003052 (Colombian numbers), A377472 (1st differences of Colombian numbers), A377473 (distinct values of A377472 in order of appearance), A377423 (= A377473 - 1).

A377473_upto(N=9, show=1)={my(o, c, d, L=List()); for(n=1+o=1, oo, is_A003052(n)||next; c++; if(!setsearch(L, d=n-o), show && printf("%d, ",[c,d]); listput(L,c); #L

a(n+1) = a(n) + 88*10^(n-1) + 1 for n = 3, 4, ..., 7 at least.

a(9) from Daniel Mondot, May 01 2025

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

Original entry on oeis.org

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

N. J. A. Sloane

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)

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.

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.
Shyam Sunder Gupta, On Some Marvellous Numbers of Kaprekar, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 9, 275-315.
D. R. Kaprekar, The Mathematics of the New Self Numbers, 1963. [annotated and scanned]
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.
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
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

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.

a003052 n = a003052_list !! (n-1)
a003052_list = filter ((== 0) . a230093) [1..]
-- Reinhard Zumkeller, Oct 11 2013, Aug 21 2011

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

nn = 525; Complement[Range[nn], Union[Table[n + Total[IntegerDigits[n]], {n, nn}]]] (* T. D. Noe, Mar 31 2013 *)

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

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

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

More terms from James Sellers, Jul 06 2000

A377423 Distinct values of the number of integers between consecutive self numbers (A163139), in order of occurrence.

Original entry on oeis.org

1, 10, 14, 27, 40, 53, 66, 79, 92, 105, 118, 100, 117, 130, 143, 156, 169, 182, 195, 208, 23, 89, 203, 220, 233, 246, 259, 272, 285, 298, 34, 78, 293, 306, 323, 336, 349, 362, 375, 388, 45, 67, 383, 396, 409, 426, 439, 452, 465, 478, 56, 473, 486, 499

Offset: 1

Daniel Mondot, Oct 27 2024

Each new value is typically found between self numbers located around 10^k, for some k.

This sequences exhibits interesting patterns, for instance, many new numbers are 13 apart.

			Between the first 2 self numbers 1 and 3, there is 1 integer. So 1 is in the sequence
The next new gap is between 9 and 20, with 10 integers, so 10 is in the sequence.
The next new gap is between 1006 and 1021, with 14 integers, so 14 is in the sequence.

Daniel Mondot, Table of n, a(n) for n = 1..10000
Daniel Mondot, First self number found after a new gap value

Cf. A003052, A163128, A163139, A225793, A230093, A230094, A377422, A377472, A377473.

a(n) = A377473(n)-1. - Daniel Mondot, Apr 17 2025

cp's OEIS Frontend

A377473 Distinct first differences of Colombian or self numbers (A377472), listed in the order they appear.

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A377474 Indices where new terms arise among first differences of Colombian or self numbers (A377472).

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

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A377423 Distinct values of the number of integers between consecutive self numbers (A163139), in order of occurrence.

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