A062028 -id:A062028 - cp's OEIS frontend

A230094 Numbers that can be expressed as (m + sum of digits of m) in exactly two ways.

101, 103, 105, 107, 109, 111, 113, 115, 117, 202, 204, 206, 208, 210, 212, 214, 216, 218, 303, 305, 307, 309, 311, 313, 315, 317, 319, 404, 406, 408, 410, 412, 414, 416, 418, 420, 505, 507, 509, 511, 513, 515, 517, 519, 521, 606, 608, 610, 612, 614, 616, 618, 620, 622, 707, 709, 711, 713, 715, 717, 719, 721, 723, 808

Offset: 1

N. J. A. Sloane, Oct 10 2013, Oct 24 2013

Numbers n such that A230093(n) = 2.
The sequence "Numbers n such that A230093(n) = 3" starts at 10^13+1 (see A230092). This implies that changing the definition of A230094 to "Numbers n such that A230093(n) >= 2" (the so-called "junction numbers") would produce a sequence which agrees with A230094 up to 10^13.
Makowski shows that the sequence of junction numbers is infinite.

			a(1) = 101 = 91 + (9+1) = 100 + (1+0+0);
a(10) = 202 = 191 + (1+9+1) = 200 + (2+0+0);
a(100) = 1106 = 1093 + (1+0+9+3) = 1102 + (1+1+0+2);
a(1000) = 10312 = 10295 + (1+0+2+9+5) = 10304 + (1+0+3+0+4).

Joshi, V. S. A note on self-numbers. Volume dedicated to the memory of V. Ramaswami Aiyar. Math. Student 39 (1971), 327--328 (1972). 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.
Makowski, Andrzej. On Kaprekar's "junction numbers''. Math. Student 34 1966 77 (1967). MR0223292 (36 #6340)
Narasinga Rao, A. On a technique for obtaining numbers with a multiplicity of generators. Math. Student 34 1966 79--84 (1967). MR0229573 (37 #5147)

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.
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]
David A. Corneth, Examples
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 [annotated and scanned]
Index entries for Colombian or self numbers and related sequences

Cf. A003052, A007953, A004207, A048528, A062028, A176995, A225793, A227915, A230092, A230093.

a230094 n = a230094_list !! (n-1)
a230094_list = filter ((== 2) . a230093) [0..]
-- Reinhard Zumkeller, Oct 11 2013

Maple
```
For Maple code see A230093.
```

Position[#, 2][[All, 1]] - 1 &@ Sort[Join[#2, Map[{#, 0} &, Complement[Range[#1], #2[[All, 1]]]] ] ][[All, -1]] & @@ {#, Tally@ Array[# + Total@ IntegerDigits@ # &, # + 1, 0]} &[10^3] (* Michael De Vlieger, Oct 28 2020, after Harvey P. Dale at A230093 *)

A258881 a(n) = n + the sum of the squared digits of n.

Original entry on oeis.org

0, 2, 6, 12, 20, 30, 42, 56, 72, 90, 11, 13, 17, 23, 31, 41, 53, 67, 83, 101, 24, 26, 30, 36, 44, 54, 66, 80, 96, 114, 39, 41, 45, 51, 59, 69, 81, 95, 111, 129, 56, 58, 62, 68, 76, 86, 98, 112, 128, 146, 75, 77, 81, 87, 95, 105, 117, 131, 147, 165, 96, 98, 102

Offset: 0

M. F. Hasler, Jul 19 2015

Carlos Rivera, Puzzle 776. Ten consecutive integers such that..., The Prime Puzzles and Problems Connection, March 2015.

Cf. A003132, A062028, A259391, A259567, A033936, A076161 (indices of primes), A329179 (indices of squares).

Total[Flatten@ {#, IntegerDigits[#]^2}] & /@ Range[0, 61] (* Michael De Vlieger, Jul 20 2015 *)
Table[n+Total[IntegerDigits[n]^2],{n,0,100}] (* Harvey P. Dale, Nov 27 2022 *)

PARI
```
A258881(n)=n+norml2(digits(n))
```

def ssd(n): return sum(int(d)**2 for d in str(n))
def a(n): return n + ssd(n)
print([a(n) for n in range(63)]) # Michael S. Branicky, Jan 30 2021

A048521 Primes expressible as the sum of an integer plus its digit sum.

Original entry on oeis.org

2, 11, 13, 17, 19, 23, 29, 37, 41, 43, 47, 59, 61, 67, 71, 73, 79, 83, 89, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 223, 227, 229, 239, 241, 251, 257, 263, 269, 271, 281, 283, 293, 307, 311, 313

Offset: 1

Patrick De Geest, May 15 1999

			a(24) = prime 113 which is 106 + (1+0+6) (or 97 + (9+7)).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for Colombian or self numbers and related sequences

Cf. A007953, A047791, A048520.
Cf. A006378, A062028, A107741.
Cf. A000040, A107740, A049084.

a048521 n = a048521_list !! (n-1)
a048521_list = map a000040 $ filter ((> 0) . a107740) [1..]
-- Reinhard Zumkeller, Sep 27 2014

t={};Do[p=Prime[n];c=0;i=1;While[iJayanta Basu, May 03 2013 *)
Union[Select[Table[n+Total[IntegerDigits[n]],{n,400}],PrimeQ]] (* Harvey P. Dale, Jul 14 2014 *)

A107740(A049084(a(n))) > 0.

Formula and also offset corrected by Reinhard Zumkeller, Sep 27 2014

A064806 a(n) = n + digital root of n.

Original entry on oeis.org

2, 4, 6, 8, 10, 12, 14, 16, 18, 11, 13, 15, 17, 19, 21, 23, 25, 27, 20, 22, 24, 26, 28, 30, 32, 34, 36, 29, 31, 33, 35, 37, 39, 41, 43, 45, 38, 40, 42, 44, 46, 48, 50, 52, 54, 47, 49, 51, 53, 55, 57, 59, 61, 63, 56, 58, 60, 62, 64, 66, 68, 70, 72, 65, 67, 69, 71

Offset: 1

Reinhard Zumkeller, Oct 21 2001

Harry J. Smith, Table of n, a(n) for n=1..1000
Index entries for Colombian or self numbers and related sequences.
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,1,-1).

Cf. A010888 (digital root of n), A062028 (sum of digits of n).

a064806 n = n + a010888 n  -- Reinhard Zumkeller, Apr 13 2013

A064806 := proc(n) return n+1 + ((n-1) mod 9): end: seq(A064806(n), n=1..100); # Nathaniel Johnston, May 04 2011

Table[n+Mod[n-1,9]+1,{n,70}] (* or *) LinearRecurrence[{1,0,0,0,0,0,0,0,1,-1},{2,4,6,8,10,12,14,16,18,11},70] (* Harvey P. Dale, Nov 19 2022 *)

a(n) = { n + (n - 1)%9 + 1 } \\ Harry J. Smith, Sep 26 2009

a(n) = n + A010888(n).
G.f.: -x*(9*x^9-2*x^8-2*x^7-2*x^6-2*x^5-2*x^4-2*x^3-2*x^2-2*x-2) / ((x-1)^2*(x^2+x+1)*(x^6+x^3+1)). - Colin Barker, Apr 05 2013
Sum_{n>=1} (-1)^(n+1)/a(n) = 263/315 + 8*Pi/(9*sqrt(3)) - log(2)/9 + (2*Pi/9)*(sec(Pi/18) - 4*cos(Pi/18)). - Amiram Eldar, May 12 2025

A066564 Numbers that when incremented by the sum of their digits produce a square.

Original entry on oeis.org

0, 2, 8, 17, 27, 38, 72, 86, 135, 161, 179, 216, 245, 275, 315, 347, 432, 467, 521, 558, 614, 662, 720, 770, 830, 882, 944, 998, 1016, 1080, 1145, 1220, 1278, 1355, 1433, 1512, 1583, 1664, 1746, 1829, 1922, 1998, 2016, 2111, 2189, 2286, 2384, 2483, 2583

Offset: 1

Amarnath Murthy, Dec 18 2001

			179 is in the sequence because 179 + sum of digits of 179 = 179 + 17 = 196 which is a perfect square. - _Indranil Ghosh_, Feb 10 2017

Indranil Ghosh, Table of n, a(n) for n = 1..11130 (terms 1..1000 from Harry J. Smith)

Cf. A062028, A066566.

[n: n in [0..3*10^3] | IsSquare(&+Intseq(n)+n)]; // Vincenzo Librandi, Jan 15 2016

Select[Range[0,2600],IntegerQ[Sqrt[#+Total[IntegerDigits[#]]]]&] (* Harvey P. Dale, May 19 2012 *)

digitsum(n) = local(s, d); s = 0; while(n>0, d = divrem(n, 10); n = d[1]; s = s+d[2]); s
a066564(m) = local(n); for(n = 0, m, if(issquare(n+digitsum(n)), print1(n, ", ")))
a066564(10000)

isok(n) = issquare(n + sumdigits(n)); \\ Michel Marcus, Jan 15 2016

More terms from Jason Earls, Dec 20 2001

A108773 Concatenation of n and the sum of the digits of n.

Original entry on oeis.org

0, 11, 22, 33, 44, 55, 66, 77, 88, 99, 101, 112, 123, 134, 145, 156, 167, 178, 189, 1910, 202, 213, 224, 235, 246, 257, 268, 279, 2810, 2911, 303, 314, 325, 336, 347, 358, 369, 3710, 3811, 3912, 404, 415, 426, 437, 448, 459, 4610, 4711, 4812, 4913, 505, 516, 527

Offset: 0

N. J. A. Sloane, Jun 26 2005

A136614(n) = A007953(a(n)) = A007953(A136613(n)). - Reinhard Zumkeller, Jan 13 2008

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A108203, A062028, A064806.

Cf. A007953.

f[n_] := FromDigits[ Join[ IntegerDigits[n], IntegerDigits[Plus @@ IntegerDigits[n]]]]; Table[ f[n], {n, 0, 52}] (* Robert G. Wilson v, Jun 28 2005 *)

a(n) = eval(concat(Str(n), Str(sumdigits(n)))); \\ Michel Marcus, Nov 12 2023

More terms from Robert G. Wilson v, Jun 28 2005

A225793 Numbers n that can be uniquely expressed as (m + sum of digits of m) for some m.

Original entry on oeis.org

2, 4, 6, 8, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 76, 77

Offset: 1

Jayanta Basu, Jul 27 2013

Subset of A176995; first member in A176995 that is not here is 101, next is 103 (cf. A230094).

A230093(a(n)) = 1. - Reinhard Zumkeller, Oct 11 2013

			100 is a member as 100 = 86 + sum of digits of (86). 101 is not a member since both 91 and 100 generate 101. Again 103 is not a member as 92 and 101 generate 103.

Joshi, V. S. A note on self-numbers. Volume dedicated to the memory of V. Ramaswami Aiyar. Math. Student 39 (1971), 327--328 (1972). MR0330032 (48 #8371)
Makowski, Andrzej. On Kaprekar's "junction numbers''. Math. Student 34 1966 77 (1967). MR0223292 (36 #6340)
Narasinga Rao, A. On a technique for obtaining numbers with a multiplicity of generators. Math. Student 34 1966 79--84 (1967). MR0229573 (37 #5147)

Jayanta Basu and Donovan Johnson, Table of n, a(n) for n = 1..10000 (first 1000 terms from Jayanta Basu)
Index entries for Colombian or self numbers and related sequences

Cf. A007953, A062028, A004207, A230093, A003052, A176995, A230094.

a225793 n = a225793_list !! (n-1)
a225793_list = filter ((== 1) . a230093) [1..]
-- Reinhard Zumkeller, Oct 11 2013

For Maple code see A230093. - N. J. A. Sloane, Oct 11 2013

co[n_] := Count[Range[n - 1], _?(# + Total[IntegerDigits[#]] == n &)]; Select[Range[100], co[#] == 1 &]
Select[Tally[Table[m+Total[IntegerDigits[m]],{m,100}]],#[[2]]==1&][[All, 1]]// Sort (* Harvey P. Dale, Aug 23 2017 *)

A065073 a(n) = prime(n) + (sum of digits of prime(n)).

Original entry on oeis.org

4, 6, 10, 14, 13, 17, 25, 29, 28, 40, 35, 47, 46, 50, 58, 61, 73, 68, 80, 79, 83, 95, 94, 106, 113, 103, 107, 115, 119, 118, 137, 136, 148, 152, 163, 158, 170, 173, 181, 184, 196, 191, 202, 206, 214, 218, 215, 230, 238, 242, 241, 253, 248, 259, 271, 274, 286

Offset: 1

Bodo Zinser, Nov 09 2001

			a(5) = 13 because p(5) = 11 and 11 + (1 + 1) = 13.

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A062028, A007953, A000040.

Cf. A047791, A048519, A048520, A006378, A107740.

a065073 = a062028 . a000040  -- Reinhard Zumkeller, Sep 27 2014

[NthPrime(n) + &+Intseq(NthPrime(n), 10): n in [1..80]]; // Vincenzo Librandi, Nov 07 2018

Table[ Prime[n] + Apply[ Plus, IntegerDigits[ Prime[n]]], {n, 1, 75} ]

forprime(p=2,300,print1(p+sumdigits(p),",")) \\ Edited by M. F. Hasler, Nov 06 2018

A065073(n)=sumdigits(n=prime(n))+n \\ M. F. Hasler, Nov 06 2018

a(n) = A062028(A000040(n)). - M. F. Hasler, Nov 06 2018

More terms from Larry Reeves (larryr(AT)acm.org) and Robert G. Wilson v, Nov 13 2001

A161351 a(n) = n + sum_of_digits(n) + product_of_digits(n).

Original entry on oeis.org

3, 6, 9, 12, 15, 18, 21, 24, 27, 11, 14, 17, 20, 23, 26, 29, 32, 35, 38, 22, 26, 30, 34, 38, 42, 46, 50, 54, 58, 33, 38, 43, 48, 53, 58, 63, 68, 73, 78, 44, 50, 56, 62, 68, 74, 80, 86, 92, 98, 55, 62, 69, 76, 83, 90, 97, 104, 111, 118, 66, 74, 82, 90, 98, 106, 114, 122, 130

Offset: 1

Claudio Meller, Jun 07 2009

a(10) = 10 + (1 + 0) + (1*0) = 11 ; a(19) = 19 + (9 + 1) + (9*1) = 38.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A007953, A007954, A011540, A062028.

Table[n+Total[IntegerDigits[n]]+Times@@IntegerDigits[n],{n,70}] (* Harvey P. Dale, Jun 07 2020 *)

a(n) = my(d=digits(n)); n + vecsum(d) + vecprod(d); \\ Michel Marcus, Nov 12 2022

from math import prod
def a(n): d = list(map(int, str(n))); return n + sum(d) + prod(d)
print([a(n) for n in range(1, 71)]) # Michael S. Branicky, Nov 20 2022

a(n) = n + A007953(n) + A007954(n). - Michel Marcus, Nov 12 2022

If n contains a digit 0 (A011540), then a(n) = A062028(n). - Bernard Schott, Nov 12 2022

A171671 Square numbers not of form m + sum of digits of m.

Original entry on oeis.org

1, 9, 64, 121, 400, 5776, 6889, 7396, 8836, 9409, 10816, 12100, 17689, 18769, 27556, 29929, 30976, 33856, 34969, 37636, 49729, 65536, 69169, 69696, 70756, 75076, 75625, 76729, 80656, 110224, 124609, 126736, 132496, 134689, 156816, 162409

Offset: 1

Zak Seidov, Dec 15 2009

We may call these numbers the self or Colombian squares. Subsequence of A003052. There are 446 such self squares < 2*10^7, 218 odd and 228 even.

Kaprekar (1963) introduced these numbers and called them self-square numbers. - N. J. A. Sloane, Oct 30 2014

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

Amiram Eldar, 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.
D. R. Kaprekar, The Mathematics of the New Self Numbers. [annotated and scanned]
Zak Seidov, Self squares.

Intersection of A000290 and A003052 (self or Colombian numbers).

Cf. A171672 (m^2 are self numbers), A062028 (a(n) = n + sum of the digits of n), A171673 (n and n^2 are self numbers), A382166.

A062028=Table[n+Total[IntegerDigits[n]],{n,0,20000000}];
se=Select[Complement[Range[0,20000000],A062028],IntegerQ[Sqrt[ # ]]&]

a(n) = A171672(n)^2. - Amiram Eldar, Mar 26 2025

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

cp's OEIS Frontend

A230094 Numbers that can be expressed as (m + sum of digits of m) in exactly two ways.

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A258881 a(n) = n + the sum of the squared digits of n.

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A048521 Primes expressible as the sum of an integer plus its digit sum.

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A064806 a(n) = n + digital root of n.

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A066564 Numbers that when incremented by the sum of their digits produce a square.

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Magma

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A108773 Concatenation of n and the sum of the digits of n.

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A225793 Numbers n that can be uniquely expressed as (m + sum of digits of m) for some m.

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