A003052 -id:A003052 - cp's OEIS frontend

A337208 Indices m of repunits R_m that are Colombian (or self) numbers.

1, 4, 6, 17, 28, 39, 50, 61, 72, 83, 94, 109, 120, 131, 142, 153, 164, 175, 186, 197, 199, 210, 221, 232, 243, 254, 265, 276, 287, 298, 300, 311, 322, 333, 344, 355, 366, 377, 388, 399, 401, 412, 423, 434, 445, 456, 467, 478, 489, 500, 502, 513, 524, 535, 546, 557

Offset: 1

David A. Corneth, Aug 19 2020

David A. Corneth, Table of n, a(n) for n = 1..10000
Shyam Sunder Gupta, On Some Marvellous Numbers of Kaprekar, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 9, 275-315.

Cf. A002275 (repunits), A003052 (Colombian (or self)), A004022 (repunit primes), A004023 (indices of repunit primes), complement of A337139.

upto(n)= {my(res = List()); for(i = 1, n, if(is(i), listput(res, i); print1(i", "))); res}
is(n) = {c = 10^n \ 9; is_A003052(c)}
is_A003052(n)={for(i=1, min(n\2, 9*#digits(n)), sumdigits(n-i)==i && return); n} \\ from M. F. Hasler, Mar 20 2011

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

Original entry on oeis.org

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

A382452 Number of self numbers <= 10^n.

Original entry on oeis.org

5, 13, 102, 983, 9784, 97786, 977787, 9777788, 97777789, 977777790, 9777777791

Offset: 1

Shyam Sunder Gupta, Mar 27 2025

Shyam Sunder Gupta, On Some Marvellous Numbers of Kaprekar, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 9, 275-315.

Cf. A003052.

A036228 a(1) = 31; a(n+1) = a(n) + sum of decimal digits of a(n).

Original entry on oeis.org

31, 35, 43, 50, 55, 65, 76, 89, 106, 113, 118, 128, 139, 152, 160, 167, 181, 191, 202, 206, 214, 221, 226, 236, 247, 260, 268, 284, 298, 317, 328, 341, 349, 365, 379, 398, 418, 431, 439, 455, 469, 488, 508, 521, 529, 545, 559, 578, 598, 620, 628, 644, 658

Offset: 1

Miklos SZABO (mike(AT)ludens.elte.hu)

Elements >= 214 can be found in A007618

D. R. Kaprekar, The Mathematics of the New Self Numbers, Privately Printed, 311 Devlali Camp, Devlali, India, 1963.

Harvey P. Dale, Table of n, a(n) for n = 1..1000
D. R. Kaprekar, The Mathematics of the New Self Numbers [annotated and scanned]
Index entries for Colombian or self numbers and related sequences

Cf. A004207, A016052, A007618, A006507, A016052, A249043.

NestList[#+Total[IntegerDigits[#]]&,31,60] (* Harvey P. Dale, Jan 30 2020 *)

Edited by Charles R Greathouse IV, Aug 02 2010

A163139 First differences of A163128.

Original entry on oeis.org

1, 1, 1, 1, 10, 10, 10, 10, 10, 10, 10, 10, 10, 1, 10, 10, 10, 10, 10, 10, 10, 10, 10, 1, 10, 10, 10, 10, 10, 10, 10, 10, 10, 1, 10, 10, 10, 10, 10, 10, 10, 10, 10, 1, 10, 10, 10, 10, 10, 10, 10, 10, 10, 1, 10, 10, 10, 10, 10, 10, 10, 10, 10, 1, 10, 10, 10, 10, 10, 10, 10, 10, 10, 1, 10

Offset: 1

Juri-Stepan Gerasimov, Jul 21 2009

First differences of self numbers, minus 1.

			a(6) = 24 - 14 = 10.

Cf. A003052, A163128.

A007953 := proc(n) add(d,d=convert(n,base,10)) ; end:
isA003052 := proc(n) for k from 1 to n do if k+A007953(k) = n then RETURN(false) ; fi; od: 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:
for n from 1 to 100 do printf("%d,",A003052(n+1)-A003052(n)-1) ; od: # R. J. Mathar, Jul 31 2009

a(n) = A163128(n+1) - A163128(n) = A003052(n+1) - A003052(n) - 1.

Missing 10's inserted by R. J. Mathar, Jul 31 2009

A171672 Numbers m with property that m^2 is not of form (k + sum of digits of k).

Original entry on oeis.org

1, 3, 8, 11, 20, 76, 83, 86, 94, 97, 104, 110, 133, 137, 166, 173, 176, 184, 187, 194, 223, 256, 263, 264, 266, 274, 275, 277, 284, 332, 353, 356, 364, 367, 396, 403, 407, 436, 443, 454, 457, 464, 504, 533, 535, 546, 587, 623, 624, 625, 634, 637, 644, 654, 673

Offset: 1

Zak Seidov, Dec 15 2009

Amiram Eldar, Table of n, a(n) for n = 1..10000
Zak Seidov, Self squares.
Zak Seidov, A171672: n^2 is safe number.

Cf. A003052 (self or Colombian numbers), A171671 (m^2 are self numbers), A062028 (a(n) = n + sum of the digits of n), A171673 (n and n^2 are self numbers).

nn=5*10^5; list=Table[n + Total[IntegerDigits[n]],{n,nn}]; Select[Sqrt[Complement[Range[nn],list]], IntegerQ[#] &] (* Jayanta Basu, May 06 2013 *)

a(n) = sqrt(A171671(n)).

Changed the word "safe" in this entry to "self". - N. J. A. Sloane, Feb 26 2017

A230865 a(n) = n + (sum of digits in base-5 representation of n).

Original entry on oeis.org

0, 2, 4, 6, 8, 6, 8, 10, 12, 14, 12, 14, 16, 18, 20, 18, 20, 22, 24, 26, 24, 26, 28, 30, 32, 26, 28, 30, 32, 34, 32, 34, 36, 38, 40, 38, 40, 42, 44, 46, 44, 46, 48, 50, 52, 50, 52, 54, 56, 58, 52, 54, 56, 58, 60, 58, 60, 62, 64, 66, 64, 66, 68, 70, 72, 70, 72, 74, 76, 78, 76, 78, 80, 82, 84, 78, 80, 82, 84, 86, 84

Offset: 0

N. J. A. Sloane, Nov 05 2013

The image of this sequence is the set of nonnegative even numbers (A005843). Joshi (1973) proved that the sequence of base-q self numbers (analogous to A003052) is the sequence of odd numbers (A005408) for all odd q. - Amiram Eldar, Nov 28 2020

V. S. Joshi, Ph.D. dissertation, Gujarat Univ., Ahmedabad (India), October, 1973.
József Sándor and Borislav Crstici, Handbook of Number theory II, Kluwer Academic Publishers, 2004, Chapter 4, p. 384-386.

Donovan Johnson, Table of n, a(n) for n = 0..10000

Cf. A053824, A230641.

Table[n + Plus @@ IntegerDigits[n, 5], {n, 0, 100}] (* Amiram Eldar, Nov 28 2020 *)

a(n) = n + A053824(n). - Amiram Eldar, Nov 28 2020

A247104 Squarefree self-numbers.

Original entry on oeis.org

3, 5, 7, 31, 42, 53, 86, 97, 110, 143, 154, 165, 187, 209, 211, 222, 233, 255, 266, 277, 299, 310, 323, 334, 345, 367, 389, 411, 413, 435, 446, 457, 479, 501, 514, 547, 569, 591, 602, 613, 615, 626, 659, 670, 681, 703, 714, 727, 749, 771, 782, 793, 815, 817

Offset: 1

Reinhard Zumkeller, Nov 18 2014

Squarefree numbers not expressible as the sum of an integer and its digit sum;

intersection of A005117 and A003052.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
D. R. Kaprekar, The Mathematics of the New Self Numbers [annotated and scanned]
Eric Weisstein's World of Mathematics, Self Number
Wikipedia, Self number

Cf. A005117, A003052, A008966, A062028, A006378 (subsequence), A249044.

a247104 n = a247104_list !! (n-1)
a247104_list = filter ((== 1) . a008966) $ tail a003052_list

nmax = 1000;
Select[Complement[Range[nmax], Union[Table[n + Total[IntegerDigits[n]], {n, 1, nmax}]]], #>1 && SquareFreeQ[#]&] (* Jean-François Alcover, Jan 08 2020, after T. D. Noe in A003052 *)

A008966(a(n)) * (1 - A230093(a(n))) = 1.

A333858 Numbers that are both Colombian and Brazilian.

Original entry on oeis.org

7, 20, 31, 42, 64, 75, 86, 108, 110, 121, 132, 143, 154, 165, 176, 187, 198, 209, 211, 222, 244, 255, 266, 288, 299, 310, 312, 323, 334, 345, 356, 378, 400, 411, 413, 424, 435, 446, 468, 490, 501, 512, 514, 525, 536, 558, 580, 591, 602, 615, 626, 637, 648, 670, 681, 692

Offset: 1

Bernard Schott, Apr 08 2020

121 is the only square of prime in this sequence.

			20 is a term because it is not of the form m + sum of digits of m for any m < 20, so 20 is Colombian and 20 = (22)_9, so 20 is also Brazilian.

Wikipédia, Nombre brésilien (in French).
Index entries for sequences related to Brazilian Numbers.
Index entries for sequences related to Colombian Numbers.

Intersection of A003052 and A125134.

brazQ[n_] := Module[{b = 2, found = False}, While[b < n - 1 && Length[Union[IntegerDigits[n, b]]] > 1, b++]; b < n - 1]; n = 700; Select[Complement[Range[n], Union @ Table[Plus @@ IntegerDigits[k] + k, {k, 1, n}]], brazQ] (* Amiram Eldar, Apr 08 2020 after T. D. Noe at A125134 *)

A336985 Colombian numbers that are not Bogotá numbers.

Original entry on oeis.org

3, 5, 7, 20, 31, 53, 86, 97, 108, 110, 121, 132, 143, 154, 165, 176, 187, 198, 209, 211, 222, 233, 244, 266, 277, 288, 299, 310, 323, 334, 345, 356, 367, 389, 400, 411, 413, 424, 435, 446, 457, 468, 479, 490, 501, 512, 514, 536, 547, 558, 569, 580, 591, 602, 613

Offset: 1

Bernard Schott, Aug 26 2020

Equivalently, numbers m that are not of the form k + sum of digits of k for any k (A003052), and that are not of the form q * product of digits of q for any q (complement of A336826).
As repunits are trivially Bogotá numbers, there are not repunits in the data.
A336983, A336984, A336986 and this sequence form a partition of the set of positive integers N*

			7 is a term because there are not k < 7  such that 7 = k + sum of digits of k, and that are not q such that 7 = q * product of digits of q.
13 is not of the form q * product of digits of q for any q <= 13, so 13 is not a Bogotá number, but 13 = 11 + (1+1) is not Colombian, hence 13 is not a term.
42 is Colombian because there does not exist m < 42 such that 42 = m + sum of digits of m, but as 42 = 21 * (2*1) is a Bogota number, 42 is not a term.

Giovanni Resta, Self or Colombian number, Numbers Aplenty.
Index to sequences related to Colombian or self numbers.

Cf. A003052 (Colombian), A176995 (not Colombian), A336826 (Bogotá numbers), A336983 (Bogotá not Colombian), A336984 (Bogotá and Colombian), this sequence (Colombian not Bogotá), A336986 (not Colombian and not Bogotá).

m = 600; Intersection[Complement[Range[m], Select[Union[Table[n + Plus @@ IntegerDigits[n], {n, 1, m}]], # <= m &]], Complement[Range[m], Select[Union[Table[n * Times @@ IntegerDigits[n], {n, 1, m}]], # <= m &]]] (* Amiram Eldar, Aug 26 2020 *)

lista(nn) = Vec(setintersect(setminus([1..nn], Set(vector(nn, k, k+sumdigits(k)))), setminus([1..nn], Set(vector(nn, k, k*vecprod(digits(k))))))); \\ Michel Marcus, Aug 26 2020

cp's OEIS Frontend

A337208 Indices m of repunits R_m that are Colombian (or self) numbers.

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

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PARI

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A382452 Number of self numbers <= 10^n.

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A036228 a(1) = 31; a(n+1) = a(n) + sum of decimal digits of a(n).

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A163139 First differences of A163128.

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Maple

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A171672 Numbers m with property that m^2 is not of form (k + sum of digits of k).

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Mathematica

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A230865 a(n) = n + (sum of digits in base-5 representation of n).

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Mathematica

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A247104 Squarefree self-numbers.

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