A004207 -id:A004207 - cp's OEIS frontend

A006507 a(n+1) = a(n) + sum of digits of a(n), with a(1)=7.

7, 14, 19, 29, 40, 44, 52, 59, 73, 83, 94, 107, 115, 122, 127, 137, 148, 161, 169, 185, 199, 218, 229, 242, 250, 257, 271, 281, 292, 305, 313, 320, 325, 335, 346, 359, 376, 392, 406, 416, 427, 440, 448, 464, 478, 497, 517, 530, 538, 554, 568

Offset: 1

N. J. A. Sloane

a(n) = A004207(n+4) for n > 11. - Reinhard Zumkeller, Oct 14 2013

Editorial Note, Popular Computing (Calabasas, CA), Vol. 4 (No. 37, Apr 1976), p. 12.
GCHQ, The GCHQ Puzzle Book, Penguin, 2016. See page 36.
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.
Jeffrey Shallit, personal communication.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
D. R. Kaprekar, The Mathematics of the New Self Numbers [annotated and scanned]
Index entries for Colombian or self numbers and related sequences

Cf. A016052, A007618, A016096.

a006507 n = a006507_list !! (n-1)
a006507_list = iterate a062028 7  -- Reinhard Zumkeller, Oct 14 2013

NestList[#+Total[IntegerDigits[#]]&,7,50] (* Harvey P. Dale, Jan 25 2021 *)

a(n) = A062028(a(n-1)) for n > 1. - Reinhard Zumkeller, Oct 14 2013

More terms from Robert G. Wilson v

A065075 Sum of digits of the sum of the preceding numbers.

Original entry on oeis.org

1, 1, 2, 4, 8, 7, 5, 10, 11, 13, 8, 7, 14, 10, 2, 4, 8, 7, 5, 10, 11, 13, 8, 16, 14, 19, 11, 13, 8, 7, 14, 10, 11, 13, 8, 7, 5, 10, 11, 13, 17, 16, 14, 10, 11, 13, 8, 16, 14, 19, 20, 13, 8, 16, 14, 19, 20, 13, 8, 16, 14, 19, 20, 22, 17, 16, 14, 19, 20, 13, 17, 16, 14, 19, 20, 13

Offset: 1

Bodo Zinser, Nov 09 2001

This sequence has the same digital roots as A004207 (a(1) = 1, a(n) = sum of digits of all previous terms) and A001370 (Sum of digits of 2^n); the digital roots sequence ends in the cycle {1 2 4 8 7 5}. - Alexandre Wajnberg, Dec 11 2005
The missing digital roots are precisely the multiples of 3. - Alexandre Wajnberg, Dec 28 2005
Conjecture: every non-multiple of 3 does appear in the sequence. - Franklin T. Adams-Watters, Jun 29 2009. See A230289. - N. J. A. Sloane, Oct 17 2013
a(n) = sum of digits of A004207(n). - N. J. A. Sloane, Oct 18 2013

			a(6) = 7 because a(1)+a(2)+a(3)+a(4)+a(5) = 16 and 7 = 1+6.

Harry J. Smith and N. J. A. Sloane, Table of n, a(n) for n = 1..10000 (the first 1000 terms were computed by Harry J. Smith)
Index entries for Colombian or self numbers and related sequences

Cf. A004207, A001370, A230289, A001651.

a065075 n = a065075_list !! (n-1)
a065075_list = 1 : 1 : f 2 where
   f x = y : f (x + y) where y = a007953 x
-- Reinhard Zumkeller, Nov 13 2014

read transforms;
sp:=1;
lprint(1,sp);
s:=sp;
for n from 2 to 10000 do
sp:=digsum(s);
lprint(n,sp);
s:=s+sp;
od:
# N. J. A. Sloane, Oct 17 2013

a065075(m) = local(a,j,s); a=1; print1(a,", "); s=a; for(j=1,m,a=sumdigits(s); print1(a,", "); s=s+a)
a065075(80)

a(1) = 1, a(2) = 1, a(n) = sum of digits of (a(1)+a(2)+...+a(n-1)).

More terms from Larry Reeves (larryr(AT)acm.org) and Klaus Brockhaus, Nov 13 2001

Edited by Franklin T. Adams-Watters, Jun 29 2009

A047892 a(1) = 2; for n > 0, a(n+1) = a(n) * sum of digits of a(n).

Original entry on oeis.org

2, 4, 16, 112, 448, 7168, 157696, 5361664, 166211584, 5651193856, 276908498944, 19383594926080, 1298700860047360, 79220752462888960, 6733763959345561600, 592571228422409420800, 45035413360103115980800

Offset: 1

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

a(n) mod 9 = A010712(n-1) for n > 1. - Reinhard Zumkeller, Sep 23 2007

Reinhard Zumkeller, Table of n, a(n) for n = 1..250

Cf. A004207.
Cf. A007953.
Cf. A047912 (start=3), A047897 (start=5), A047898 (start=6), A047899 (start=7), A047900 (start=8), A047901 (start=9), A047902 (start=11).

a047892 n = a047892_list !! (n-1)
a047892_list = iterate a057147 2  -- Reinhard Zumkeller, Mar 19 2014

NestList[# Total[IntegerDigits[#]]&,2,20] (* Harvey P. Dale, Jul 18 2011 *)

a(n+1) = A057147(a(n)). - Reinhard Zumkeller, Mar 19 2014

Offset changed by Reinhard Zumkeller, Mar 19 2014

A230099 a(n) = n + (product of digits of n).

Original entry on oeis.org

0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 20, 23, 26, 29, 32, 35, 38, 41, 44, 47, 30, 34, 38, 42, 46, 50, 54, 58, 62, 66, 40, 45, 50, 55, 60, 65, 70, 75, 80, 85, 50, 56, 62, 68, 74, 80, 86, 92, 98, 104, 60, 67, 74, 81, 88, 95, 102, 109, 116, 123, 70, 78, 86, 94, 102, 110, 118, 126

Offset: 0

N. J. A. Sloane, Oct 12 2013

A230099, A063114, A098736, A230101 are analogs of A092391 and A062028.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Index entries for Colombian or self numbers and related sequences

Cf. A007954, A011540, A092391, A062028, A004207, A230093, A003052, A176995.

Cf. also A063114, A098736, A230101, A230103, A230104, A230105, A062028, A230102, A063108.

a230099 n = a007954 n + n  -- Reinhard Zumkeller, Oct 13 2013

with transforms; [seq(n+digprod(n), n=0..200)];

a(n) = if (n, n + vecprod(digits(n)), 0); \\ Michel Marcus, Dec 18 2018

from math import prod
def a(n): return n + prod(map(int, str(n)))
print([a(n) for n in range(78)]) # Michael S. Branicky, Jan 09 2023

a(n) = n iff n contains a digit 0 (A011540). - Bernard Schott, Jul 31 2023

A010065 a(n+1) = a(n) + sum of digits in base 4 representation of a(n), with a(0) = 1.

Original entry on oeis.org

1, 2, 4, 5, 7, 11, 16, 17, 19, 23, 28, 32, 34, 38, 43, 50, 55, 62, 70, 74, 79, 86, 91, 98, 103, 110, 118, 125, 133, 137, 142, 149, 154, 161, 166, 173, 181, 188, 196, 200, 205, 212, 217, 224, 229, 236, 244, 251, 262, 266, 271, 278, 283, 290, 295

Offset: 0

Leonid Broukhis

S. R. Finch, Mathematical Constants, Cambridge, 2003, Section 2.24.

Donovan Johnson, Table of n, a(n) for n = 0..10000
Index entries for Colombian or self numbers and related sequences

Cf. A003052, A010061, A010062, A010064, A004207.

Related base-4 sequences: A053737, A230631, A230632, A010064, A230633, A230634, A230635, A230636, A230637, A230638, A010065 (trajectory of 1)

a010065 n = a010065_list !! n
a010065_list = iterate a230631 1  -- Reinhard Zumkeller, Mar 20 2015

a(n+1) = A230631(a(n)). - Reinhard Zumkeller, Mar 20 2015

More terms from Neven Juric, Apr 11 2008

A016096 a(n+1) = a(n) + sum of its digits, with a(1) = 9.

Original entry on oeis.org

9, 18, 27, 36, 45, 54, 63, 72, 81, 90, 99, 117, 126, 135, 144, 153, 162, 171, 180, 189, 207, 216, 225, 234, 243, 252, 261, 270, 279, 297, 315, 324, 333, 342, 351, 360, 369, 387, 405, 414, 423, 432, 441, 450, 459, 477, 495, 513, 522, 531, 540

Offset: 1

Robert G. Wilson v

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.

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

a016096 n = a016096_list !! (n-1)
a016096_list = iterate a062028 9  -- Reinhard Zumkeller, Oct 14 2013

from itertools import islice
def A016096_gen(): # generator of terms
    a = 9
    while True:
        yield a
        a += sum(int(d) for d in str(a))
A016096_list = list(islice(A016096_gen(),20)) # Chai Wah Wu, Mar 29 2022

a(n) = A062028(a(n-1)) for n > 1. - Reinhard Zumkeller, Oct 14 2013

A007612 a(n+1) = a(n) + digital root (A010888) of a(n).

Original entry on oeis.org

1, 2, 4, 8, 16, 23, 28, 29, 31, 35, 43, 50, 55, 56, 58, 62, 70, 77, 82, 83, 85, 89, 97, 104, 109, 110, 112, 116, 124, 131, 136, 137, 139, 143, 151, 158, 163, 164, 166, 170, 178, 185, 190, 191, 193, 197, 205, 212, 217, 218, 220, 224, 232, 239, 244, 245, 247, 251

Offset: 1

N. J. A. Sloane, Robert G. Wilson v, Mira Bernstein

Take m, a natural number. If m == 1 (mod 6), then for every n a(m)*a(n) is in A007612. - Ivan N. Ianakiev, May 08 2013

J. Roberts, Lure of the Integers, Math. Assoc. America, 1992, p. 65.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Digital Root.

Cf. A004207, A010888, A029898, A064806.

a007612 n = a007612_list !! (n-1)
a007612_list = iterate a064806 1  -- Reinhard Zumkeller, Apr 13 2013

A007612 := proc(n) option remember: if(n=1)then return 1: fi: return procname(n-1) + ((procname(n-1)-1) mod 9) + 1: end: seq(A007612(n), n=1..100); # Nathaniel Johnston, May 04 2011

dr[n_]:=NestWhile[Total[IntegerDigits[#]]&,n,#>9&]; NestList[#+dr[#]&, 1,60] (* Harvey P. Dale, Sep 24 2011 *)
NestList[#+Mod[#,9]&,1,60] (* Harvey P. Dale, Sep 14 2016 *)

first(n)=my(v=vector(n)); v[1]=1; for(k=2,n, v[k]=v[k-1]+v[k-1]%9); v \\ Charles R Greathouse IV, Jun 25 2017

a(n)=n\6*27 + [-4,1,2,4,8,16][n%6+1] \\ Charles R Greathouse IV, Jun 25 2017

a(1) = 1, a(n+1) = a(n) + a(n) mod 9. - Reinhard Zumkeller, Mar 23 2003
First differences are [1,2,4,8,7,5] repeated. - M. F. Hasler, Sep 15 2009; corrected by John Keith, Aug 17 2022
n == 1, 2, 4, 8, 16, or 23 (mod 27). - Dean Hickerson, Mar 25 2003
Limit_{n->oo} a(n)/n = 9/2; A029898(n) = a(n+1) - a(n) = A010888(a(n)). - Reinhard Zumkeller, Feb 27 2006
a(6n+1)=27n+1, a(6n+2)=27n+2, a(6n+3)=27n+4, a(6n+4)=27n+8, a(6n+5)=27n+16, a(6n+6)=27n+23. - Franklin T. Adams-Watters, Mar 13 2006
G.f.: (1+4*x^4+3*x^3+x^2)/((x+1)*(x^2-x+1)*(x-1)^2). - Maksym Voznyy (voznyy(AT)mail.ru), Aug 10 2009
a(n+1) = A064806(a(n)). - Reinhard Zumkeller, Apr 13 2013

A033298 a(n+1) = a(n) + sum of digits of a(n)^2, with a(1) = 1.

Original entry on oeis.org

1, 2, 6, 15, 24, 42, 60, 69, 87, 114, 141, 168, 186, 213, 240, 258, 285, 303, 330, 348, 357, 384, 411, 438, 465, 483, 510, 519, 546, 573, 600, 609, 636, 663, 699, 726, 753, 780, 798, 825, 852, 879, 906, 933, 969, 1005, 1014, 1041, 1068, 1086

Offset: 1

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

			a(6) = 42 as a(5) = 24 giving a(6) = 24 + (sum of digits of 24^2 = 576) = 24 + 5 + 7 + 6 = 42. - _David A. Corneth_, Jun 26 2022

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

Cf. A047892, A004207, A007953.

Partial sums of A139417.

NestList[#+Total[IntegerDigits[#^2]]&,1,50] (* Harvey P. Dale, Sep 21 2023 *)

first(n) = {n=max(n,1); my(res=vector(n)); res[1] = 1; for(i = 2, n, res[i] = res[i-1] + sumdigits(res[i-1]^2)); res} \\ David A. Corneth, Jun 26 2022

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

Original entry on oeis.org

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 *)

A309261 The sum of the number of times each unique digit in the previous number occurred in the numbers before that with a(1) = 0.

Original entry on oeis.org

0, 0, 1, 0, 2, 0, 3, 0, 4, 0, 5, 0, 6, 0, 7, 0, 8, 0, 9, 0, 10, 12, 3, 1, 3, 2, 2, 3, 3, 4, 1, 4, 2, 4, 3, 5, 1, 5, 2, 5, 3, 6, 1, 6, 2, 6, 3, 7, 1, 7, 2, 7, 3, 8, 1, 8, 2, 8, 3, 9, 1, 9, 2, 9, 3, 10, 22, 10, 24, 16, 16, 18, 18, 20, 27, 18, 22, 15, 21, 35, 16, 25, 24, 24

Offset: 1

Nic Tomlinson, Jul 19 2019

			For n=2, the previous term a(1) is 0 by definition and the digit 0 has never been seen before so a(2) is 0.
For n=3, a(2) is 0, the digit 0 has been seen once before so a(3) is 1.
For n=22, a(21) is 10, digit 1 has been seen once before a(21) and digit 0 has been seen eleven times, so a(22) = 12.
For n=68, a(67) is 22, digit 2 has been seen ten times before, we only consider unique digits of a(n-1), so a(68) = 10.

Rémy Sigrist, Table of n, a(n) for n = 1..20000
Rémy Sigrist, Density plot of the first million terms

Cf. A004207.

{ f=vector(base=10); for(n=1, 84, if(n==1, v=0, d=if(v, digits(v, base), [0]); s=Set(d); v=sum(i=1, #s, f[1+s[i]]); apply(t -> f[1+t]++, d)); print1(v ", ")) } \\ Rémy Sigrist, Jul 24 2019

from collections import Counter
from itertools import count, islice
def agen(): # generator of terms
    an, c = 0, Counter()
    while True:
        yield an
        s = str(an)
        an = sum(c[d] for d in set(s))
        c.update(s)
print(list(islice(agen(), 84))) # Michael S. Branicky, Mar 24 2025

cp's OEIS Frontend

A006507 a(n+1) = a(n) + sum of digits of a(n), with a(1)=7.

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A065075 Sum of digits of the sum of the preceding numbers.

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A047892 a(1) = 2; for n > 0, a(n+1) = a(n) * sum of digits of a(n).

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A230099 a(n) = n + (product of digits of n).

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A010065 a(n+1) = a(n) + sum of digits in base 4 representation of a(n), with a(0) = 1.

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A016096 a(n+1) = a(n) + sum of its digits, with a(1) = 9.

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A007612 a(n+1) = a(n) + digital root (A010888) of a(n).

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