A062028 -id:A062028 - cp's OEIS frontend

A232490 Numbers k such that 10^k is of the form m + sum of digits of m.

1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 25, 27, 28, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 45, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 58, 59, 60, 61, 62, 63, 64, 65, 66, 68, 69, 70, 71, 72, 73, 74, 75, 76, 78, 79, 80, 81, 82, 83, 84, 85, 86, 88, 89, 90, 91, 92, 93

Offset: 1

N. J. A. Sloane, Dec 01 2013

M. Gardner, Time Travel and Other Mathematical Bewilderments. Freeman, NY, 1988, pp. 115-117 and 122.
D. R. Kaprekar, Puzzles of the Self-Numbers. 311 Devlali Camp, Devlali, India, 1959.
D. R. Kaprekar, The Mathematics of the New Self Numbers (Part V). 311 Devlali Camp, Devlali, India, 1967.

Donovan Johnson, Table of n, a(n) for n = 1..10000
T. Trotter, Charlene Numbers
Index entries for Colombian or self numbers and related sequences

Cf. A062028, A232489, A232491.

Terms a(16) onward from Max Alekseyev, Dec 02 2013

A107741 Smallest number m such that prime(n) = m + (digit sum of m), a(n)=0 if no such m exists.

Original entry on oeis.org

1, 0, 0, 0, 10, 11, 13, 14, 16, 19, 0, 32, 34, 35, 37, 0, 52, 53, 56, 58, 59, 71, 73, 76, 0, 91, 92, 94, 95, 97, 122, 124, 127, 128, 142, 143, 146, 149, 160, 163, 166, 167, 181, 182, 184, 185, 0, 215, 217, 218, 0, 232, 233, 238, 250, 253, 256, 257, 0, 271

Offset: 1

Reinhard Zumkeller, May 23 2005

If a(n)>0 then: A000040(n)=A062028(a(n)) and A107740(n)>0.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A047791, A048521.

Cf. A000040, A055642, A062028.

a107741 n = if null ms then 0 else head ms  where
   ms = [m | let p = a000040 n,
             m <- [max 0 (p - 9 * a055642 p) .. p - 1], a062028 m == p]
-- Reinhard Zumkeller, Sep 27 2014

Data error corrected by Reinhard Zumkeller, Sep 27 2014

A107743 Numbers m such that m+(digit sum of m) is a composite number.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 8, 9, 12, 15, 17, 18, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 33, 36, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 54, 55, 57, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 72, 74, 75, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 93

Offset: 1

Reinhard Zumkeller, May 23 2005

Complement of A047791.

			A062028(21)=21+(2+1)=24=2*2*2*3, therefore 21 is a term.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A007953, A062028.

Cf. A010051.

a107743 n = a107743_list !! (n-1)
a107743_list = filter ((== 0) . a010051' . a062028) [1..]
-- Reinhard Zumkeller, Sep 27 2014

Select[Range[100],CompositeQ[#+Total[IntegerDigits[#]]]&] (* Harvey P. Dale, Oct 12 2016 *)

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

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.

A248110 Table read by rows: n-th row contains the q successors of n, where q = A007953(n), the digit sum of n in decimal representation.

Original entry on oeis.org

2, 3, 4, 4, 5, 6, 5, 6, 7, 8, 6, 7, 8, 9, 10, 7, 8, 9, 10, 11, 12, 8, 9, 10, 11, 12, 13, 14, 9, 10, 11, 12, 13, 14, 15, 16, 10, 11, 12, 13, 14, 15, 16, 17, 18, 11, 12, 13, 13, 14, 15, 14, 15, 16, 17, 15, 16, 17, 18, 19, 16, 17, 18, 19, 20, 21, 17, 18, 19, 20

Offset: 1

Reinhard Zumkeller, Oct 01 2014

First 9 rows coincide with triangle A108872;

T(n,1) = n + 1; T(n,A007953(n)) = n + A007953(n) = A062028(n).

			.   n |   T(n,*)                                 | A007953(n)
.  ---+------------------------------------------+-----------
.   1 |    2                                     |       1
.   2 |    3,  4                                 |       2
.   3 |    4,  5,  6                             |       3
.   4 |    5,  6,  7,  8                         |       4
.   5 |    6,  7,  8,  9, 10                     |       5
.   6 |    7,  8,  9, 10, 11, 12                 |       6
.   7 |    8,  9, 10, 11, 12, 13, 14             |       7
.   8 |    9, 10, 11, 12, 13, 14, 15, 16         |       8
.   9 |   10, 11, 12, 13, 14, 15, 16, 17, 18     |       9
.  10 |   11                                     |       1
.  11 |   12, 13                                 |       2
.  12 |   13, 14, 15                             |       3
.  13 |   14, 15, 16, 17                         |       4
.  14 |   15, 16, 17, 18, 19                     |       5
.  15 |   16, 17, 18, 19, 20, 21                 |       6
.  16 |   17, 18, 19, 20, 21, 22, 23             |       7
.  17 |   18, 19, 20, 21, 22, 23, 24, 25         |       8
.  18 |   19, 20, 21, 22, 23, 24, 25, 26, 27     |       9
.  19 |   20, 21, 22, 23, 24, 25, 26, 27, 28, 29 |      10
.  20 |   21, 22                                 |       2

Reinhard Zumkeller, Rows n = 1..1000 of triangle, flattened

Cf. A007953 (row lengths), A062028, A108872.

a248110 n k = a248110_tabf !! (n-1) !! (k-1)
a248110_row n = a248110_tabf !! (n-1)
a248110_tabf = map (\x -> [x + 1 .. x + a007953 x]) [1 ..]

A350229 a(n) is the sum of n and the balanced ternary digits in n.

Original entry on oeis.org

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

Offset: 0

Rémy Sigrist, Jan 09 2022

The image of this sequence is the set of nonnegative even numbers (A005843).

			For n = 42:
- the balanced ternary representation of 42 is "1TTT0",
- so a(42) = 42 + 1 - 1 - 1 - 1 + 0 = 40.

See A062028, A092391, A230641 for similar sequences.

Cf. A005843, A065363, A174658 (fixed points).

Array[# + Total[If[First@ # == 0, Rest@ #, #] &[Prepend[IntegerDigits[#, 3], 0] //. {x___, y_, k_ /; k > 1, z___} :> {x, y + 1, k - 3, z}]] &, 70, 0] (* Michael De Vlieger, Jan 15 2022 *)

a(n) = my (v=n, d); while (n, n=(n-d=[0,1,-1][1+n%3])/3; v+=d); v

a(n) = n + A065363(n).

a(n) = n iff n belongs to A174658.

A377422 Numbers that can be expressed as (m + sum of digits of m) in exactly four ways.

Original entry on oeis.org

1000000000000000000000102, 1000000000000000000000104, 1000000000000000000000106, 1000000000000000000000108, 1000000000000000000000110, 1000000000000000000000112, 1000000000000000000000114, 2000000000000000000000103, 2000000000000000000000105, 2000000000000000000000107, 2000000000000000000000109

Offset: 1

Daniel Mondot, Oct 27 2024

Numbers k such that A230093(k) = 0 give A003052, the Self or Colombian numbers.
Numbers k such that A230093(k) = 1 give A225793.
Numbers k such that A230093(k) = 2 give A230094.
Numbers k such that A230093(k) = 3 give A230100.
Numbers k such that A230093(k) = 4 give this sequence.

			There are exactly four numbers, 999999999999999999999894, 999999999999999999999903, 1000000000000000000000092, and 1000000000000000000000101, whose image under n->f(n) is 1000000000000000000000104, so 1000000000000000000000104 is a member of the sequence.

Daniel Mondot, 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.

Cf. A230100, A006064, A062028, A230093.

Corrected by Daniel Mondot, Apr 17 2025

A084661 Numbers k such that k + sum_of_digits(k) is a cube.

Original entry on oeis.org

4, 18, 121, 198, 207, 329, 720, 977, 1318, 2183, 2731, 3357, 4082, 4891, 4900, 5814, 6836, 7969, 9243, 10634, 12154, 13797, 13806, 15611, 17554, 19656, 21929, 24367, 26973, 29759, 32746, 39281, 42853, 46629, 54850, 59292, 59301, 63968, 68890, 74070, 79475

Offset: 1

Zak Seidov, Jun 29 2003

A066564 lists numbers k such that k + sum_of_digits(k) is a square.

			a(3)=121 because 121 + (1 + 2 + 1) = 125 = 5^3.

Harvey P. Dale, Table of n, a(n) for n = 1..250

Cf. A062028, A066564.

Select[Range[80000],IntegerQ[Surd[#+Total[IntegerDigits[#]],3]]&] (* Harvey P. Dale, Sep 13 2018 *)

isok(n) = ispower(n + sumdigits(n), 3); \\ Michel Marcus, Oct 09 2013

More terms from Michel Marcus, Oct 04 2013

A108638 Semiprime plus its digits is semiprime.

Original entry on oeis.org

15, 22, 26, 33, 38, 39, 49, 51, 55, 57, 74, 77, 115, 123, 129, 134, 145, 155, 161, 169, 178, 187, 202, 206, 213, 214, 221, 237, 254, 265, 274, 278, 291, 299, 301, 303, 309, 321, 327, 335, 361, 371, 377, 381, 382, 386, 411, 437, 445, 466, 478, 485, 497, 505

Offset: 1

Zak Seidov, Jun 14 2005

Members k of A001358 such that A062028(k) is in A001358. - Robert Israel, Oct 01 2024
Surprisingly there are only three(?) semiprimes sp, 10,14,15, such that sp minus its digits is semiprime.
That is because n - (sum of its digits) = A066568(n) is divisible by 9. - Robert Israel, Oct 01 2024

			15=3*5 and 15+1+5=21=3*7.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A001358, A062028, A066568.

filter:= n -> numtheory:-bigomega(n) = 2 and numtheory:-bigomega(n+convert(convert(n,base,10),`+`))=2:
select(filter, [$1..1000]); # Robert Israel, Oct 01 2024

Select[Range[500], Plus@@Last/@FactorInteger[ # ]==Plus@@Last/@FactorInteger[ #+Plus@@IntegerDigits[ # ]]==2&]

cp's OEIS Frontend

A232490 Numbers k such that 10^k is of the form m + sum of digits of m.

Views

Author

Keywords

References

Links

Crossrefs

Extensions

A107741 Smallest number m such that prime(n) = m + (digit sum of m), a(n)=0 if no such m exists.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Extensions

A107743 Numbers m such that m+(digit sum of m) is a composite number.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Mathematica

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

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A247104 Squarefree self-numbers.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Mathematica

Formula

A248110 Table read by rows: n-th row contains the q successors of n, where q = A007953(n), the digit sum of n in decimal representation.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

A350229 a(n) is the sum of n and the balanced ternary digits in n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI

Formula

A377422 Numbers that can be expressed as (m + sum of digits of m) in exactly four ways.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs