A004207 -id:A004207 - cp's OEIS frontend

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

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 *)

A229168 Define a sequence of real numbers by b(1)=2, b(n+1) = b(n) + log_2(b(n)); a(n) = smallest i such that b(i) >= 2^n.

Original entry on oeis.org

1, 3, 5, 7, 11, 17, 27, 44, 74, 127, 225, 402, 728, 1333, 2459, 4566, 8525, 15993, 30122, 56936, 107953, 205253, 391223, 747369, 1430648, 2743721, 5270959, 10141978, 19542806, 37708232, 72849931, 140905791, 272836175, 528832794, 1026008203, 1992390617

Offset: 1

N. J. A. Sloane, Sep 27 2013

nonn

			The initial terms of the b(n) sequence are approximately
2, 3.00000000000000000000000, 4.58496250072115618145375, 6.78187243514238888864578, 9.54355608312733448665509, 12.7980830210090262451102, 16.4759388461842196480290, 20.5182276175427023220954, 24.8770618274970204646817, 29.5138060245244394221195, 34.3971240984210783617324, ...
b(5) is the first term >= 8, so a(3) = 5.

Index entries for Colombian or self numbers and related sequences

Cf. A229169, A229170, A010062, A229167, A155921, A229177; also A004207, A229171, A229172, A229173.

# A229168, A229169, A229170.
Digits:=24;
log2:=evalf(log(2));
lis:=[2]; a:=2;
t1:=[1]; l:=2;
for i from 2 to 128 do
a:=evalf(a+log(a)/log2);
if a >= 2^l then
l:=l+1; t1:=[op(t1),i]; fi;
lis:=[op(lis),a];
od:
lis;
map(floor,lis);
map(ceil,lis);
t1;

n=1; p2=2^n; m=2; lg2=log(2); for(i=1, 1992390617, if(m>=p2, print(n " " i); n++; p2=2^n); m=m+log(m)/lg2) /* Donovan Johnson, Oct 04 2013 */

a(11)-a(36) from Donovan Johnson, Oct 04 2013

A219675 Starting with a(0)=0, a(n) = 1 + the sum of the digital sums of a(0) through a(n-1).

Original entry on oeis.org

0, 1, 2, 4, 8, 16, 23, 28, 38, 49, 62, 70, 77, 91, 101, 103, 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: 0

Bob Selcoe, Nov 17 2014

Almost identical to A004207, only difference being a(0). - Yuval Filmus, Apr 22 2016.

			a(7) = 28 because (0+1+2+4+8+1+6+2+3) + 1 = 28.

Eric Angelini, a(n) > cumulative sum of digits, Seqfan, Nov. 11, 2014.

Cf. A004207 (essentially the same), A007953 (sum of digits), A244510 (related).

a219675[n_Integer] := Module[{f}, f[0] = 0; f[k_] := 1 + Sum[Plus @@ IntegerDigits[f[i]], {i, 0, k - 1}]; f[n]]; a219675/@Range[40] (* Michael De Vlieger, Nov 17 2014 *)

lista(nn) = {v = vector(nn); for (n=2, nn, v[n] = 1 + sum(k=1, n-1, sumdigits(v[k])););v;} \\ Michel Marcus, Nov 17 2014

A219675_upto(n)=vector(n,i,n=if(i<3, i-1, n+sumdigits(n))) \\ M. F. Hasler, Oct 30 2024

a(n) = Sum_{k=0..n-1} digsum(a(k)) + 1.

a(n) = a(n-1) + digsum(a(n-1)).

More terms from Michel Marcus, Nov 17 2014

A229171 Define a sequence of real numbers by b(1)=e, b(n+1) = b(n) + log(b(n)); a(n) = smallest i such that b(i) >= e^n.

Original entry on oeis.org

1, 5, 10, 20, 41, 86, 192, 441, 1039, 2493, 6072, 14960, 37199, 93193, 234957, 595562, 1516639, 3877905, 9950908, 25615654, 66127187, 171144672, 443966371, 1154115393, 3005950908

Offset: 1

N. J. A. Sloane, Sep 27 2013

			The initial terms of the b(n) sequence are approximately
2.71828182845904523536029, 3.71828182845904523536029, 5.03154351597726806940929, 6.64727031503970856301384, 8.54147660649653209023621, 10.6864105040926911986276, 13.0553833920216929230460, 15.6245839611886549261305, 18.3734295299727029212384, 21.2843351036624388705641, 24.3423064646657059114213, 27.5345223079930416816192, 30.8499628820185220765989, ...
b(5) is the first term >= e^2, so a(2) = 5.

Index entries for Colombian or self numbers and related sequences

Cf. A229172, A229173, A229175; A229168, A229169, A229170, A010062, A229167; also A004207.

# A229171, A229172, A229173.
Digits:=24;
e:=evalf(exp(1));
lis:=[e]; a:=e;
t1:=[1]; l:=2;
for i from 2 to 128 do
a:=evalf(a+log(a));
if a >= e^l then
l:=l+1; t1:=[op(t1),i]; fi;
lis:=[op(lis),a];
od:
lis;
map(floor,lis);
map(ceil,lis);
t1;

n=1; m=exp(1); mn=m^n; for(i=1, 3005950908, if(m>=mn, print(n " " i); n++; mn=exp(1)^n); m=m+log(m)) /* Donovan Johnson, Oct 04 2013 */

a(7)-a(25) from Donovan Johnson, Oct 04 2013

A286660 a(n) = a(n-1) + sum of base-100 digits of a(n-1), a(0) = 1.

Original entry on oeis.org

1, 2, 4, 8, 16, 32, 64, 128, 157, 215, 232, 266, 334, 371, 445, 494, 592, 689, 784, 875, 958, 1025, 1060, 1130, 1171, 1253, 1318, 1349, 1411, 1436, 1486, 1586, 1687, 1790, 1897, 2012, 2044, 2108, 2137, 2195, 2311, 2345, 2413, 2450, 2524, 2573, 2671, 2768, 2863, 2954, 3037, 3104, 3139

Offset: 0

Peter Weiss, May 12 2017

			a(7) = 128 = 1 * 100^1 + 28 * 100^0. The sum of digits of a(8 - 1) = 128 in base 100 is therefore 1 + 28 = 29. a(8) = a(7) + the sum of digits of a(7) in base 100 is therefore 128 + 29 = 157.

Robert Israel, Table of n, a(n) for n = 0..10000

Cf. A004207, A010062.

g:= n -> n+convert(convert(n,base,100),`+`):
A[0]:= 1:
for n from 1 to 100 do A[n]:= g(A[n-1]) od:
seq(A[i],i=0..100); # Robert Israel, May 22 2017

a[0] = 1; a[n_] := a[n] = a[n-1] + Total[IntegerDigits[a[n-1], 100]];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, May 21 2017 *)
NestList[#+Total[IntegerDigits[#,100]]&,1,60] (* Harvey P. Dale, May 26 2019 *)

a(n) = if(n < 8, return(1<<(n-1))); my(r = cr = 128); for(i=8, n, while(cr > 0, r += cr % 100; cr \= 100); cr = r); r \\ David A. Corneth, May 15 2017

A326834 a(0) = 0; a(1) = 0; for n > 0, a(n) = the sum of the number of times each digit in a(n-1) has occurred from a(0) to a(n-2) inclusive.

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, 20, 25, 17, 15, 17, 18, 18, 20, 28, 21, 32, 28, 25, 25, 27

Offset: 0

Scott R. Shannon, Oct 20 2019

This sequence sums the previous digits the same way as A309261, except that here digits in a(n-1) are not considered unique, so each digit in a(n-1) is summed regardless of the number of times it appears in a(n-1). This leads to this sequence being the same as A309261 up to a(66) = 22, after which they diverge.

			a(2) = 1, as a(1) = 0, and '0' has occurred one previous time in the sequence before a(1).
a(62) = 2, as a(61) = 9, and '9' has occurred two previous times. '2' has now occurred 10 times in the sequence.
a(66) = 22, as a(65) = 10, and '1' has occurred ten previous times, and '0' has occurred twelve previous times, and 10 + 12 = 22.
a(67) = 20, as a(66) = 22, and '2' has occurred ten previous times, and '2' has occurred ten previous times, and 10 + 10 = 20.

Scott R. Shannon, Table of n, a(n) for n = 0..19999
Scott R. Shannon, Plot of a(n) for n = 0..10^6 [This is a dramatic graph - _N. J. A. Sloane_, Oct 21 2019]

Cf. A004207, A309261.

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 s)
        c.update(s)
print(list(islice(agen(), 82))) # Michael S. Branicky, Mar 24 2025

A084228 a(1)=1, a(2)=2; thereafter a(n) = sum of digits of (a(1)+a(2)+a(3)+...+a(n-1)).

Original entry on oeis.org

1, 2, 3, 6, 3, 6, 3, 6, 3, 6, 12, 6, 12, 15, 12, 15, 3, 6, 3, 6, 12, 6, 12, 15, 12, 15, 3, 6, 3, 6, 12, 6, 12, 15, 12, 15, 12, 6, 12, 6, 12, 15, 12, 15, 12, 15, 12, 6, 12, 15, 12, 15, 12, 15, 12, 15, 12, 15, 12, 15, 21, 15, 12, 15, 12, 15, 21, 24, 12, 15, 12, 15, 21, 24, 12, 15, 21, 15

Offset: 1

Benoit Cloitre, Jun 21 2003

a(n) == 3 or 6 (mod 9) n>2.
a(n) = 3 for n in A084229.
a(n) = 6 for n = 4, 6, 8, 10, 12, 18, 20, 22, 28, 30, 32, 38, 40, 48, 86, 88, 90, 96, 98, 100, 106, 108, 116, 160, 162, 168, 170, 178, ..., 17630. - Robert G. Wilson v, Jun 27 2014

Robert G. Wilson v, Table of n, a(n) for n = 1..10000
Index entries for Colombian or self numbers and related sequences

Cf. A065075, A016052, A230286, A230287, A004207, A084229.

a084228 n = a084228_list !! (n-1)
a084228_list = 1 : 2 : f 3 where
   f x = y : f (x + y) where y = a007953 x
-- Reinhard Zumkeller, Nov 13 2014

a[1] = 1; a[2] = 2; a[n_] := a[n] = Sum[ Total@ IntegerDigits@ a@ i, {i, n - 1}]; Array[ Total@ IntegerDigits@ a@# &, 78] (* Robert G. Wilson v, Jun 27 2014 *)
nxt[{t_,a_}]:=Module[{c=Total[IntegerDigits[t]]},{t+c,c}]; Join[{1},NestList[nxt,{3,2},80][[;;,2]]] (* Harvey P. Dale, Jul 13 2023 *)

sumdig(n)=sum(k=0,ceil(log(n)/log(10)),floor(n/10^k)%10)
an=vector(10000); a(n)=if(n<0,0,an[n])
an[1]=1; an[2]=2; for(n=3,300,an[n]=sumdig(sum(k=1,n-1,a(k))))

A154770 a(n+1) = A154771(a(n)) = sum of all distinct "valid substrings" of a(n); a(1)=10 (least nontrivial choice).

Original entry on oeis.org

10, 11, 12, 15, 21, 24, 30, 33, 36, 45, 54, 63, 72, 81, 90, 99, 108, 127, 176, 283, 407, 458, 578, 733, 849, 1003, 1117, 1381, 2044, 2318, 2953, 4397, 5435, 6557, 7964, 9989, 12279, 16572, 26426, 36970, 49970, 67738, 84716, 100181, 111599, 139413, 209606

Offset: 1

Eric Angelini and M. F. Hasler, Jan 16 2009

"Valid substrings" means all numbers that appear as substring of n (written in decimal system). Starting values < 10 would yield a constant sequence.

			a(1) = 10 has { 0, 1, 10 } as distinct substrings,
a(2) = 0+1+10 = 11 has { 1, 11 } as distinct substrings,
a(3) = 1+11 = 12 has { 1, 2, 12 } as distinct substrings,
a(4) = 1+2+12 = 15 has { 1, 5, 15 } as distinct substrings.

E. Angelini, Sum of all valid substrings (VS), SeqFan list, Jan 16 2009

Cf. A004207, A154771.

A154770( n=2, a=10 ) = { local(d); while( n--, a=vecsort(concat(vector(d=#Str(a),i,vector(d,j,a%10^j)+(d--&!a\=10))),NULL,8);a*=vector(#a,i,1)~); a }
print1(a=10);for(i=1,50,print1("," a=A154770(,a)))

The word "distinct" added to definition Jan 19 2009 at the suggestion of Hugo van der Sanden.

A169732 a(1) = 1000; for n>1, a(n) = a(n-1) - digitsum(a(n-1)).

Original entry on oeis.org

1000, 999, 972, 954, 936, 918, 900, 891, 873, 855, 837, 819, 801, 792, 774, 756, 738, 720, 711, 702, 693, 675, 657, 639, 621, 612, 603, 594, 576, 558, 540, 531, 522, 513, 504, 495, 477, 459, 441, 432, 423, 414, 405, 396, 378, 360, 351, 342, 333, 324, 315, 306, 297, 279, 261, 252, 243, 234, 225, 216, 207, 198, 180, 171, 162, 153, 144, 135, 126, 117, 108, 99, 81, 72, 63, 54, 45, 36, 27, 18, 9, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

N. J. A. Sloane, May 01 2010, based on a suggestion from Chris Cole

Cf. A169733, A000045, A004207, A007618, A065075, A010077, A065076, A065124, A169734, A169735.

f:=proc(n) global S; option remember; if n=1 then RETURN(S) else RETURN(f(n-1)-digsum(f(n-1))); fi; end; S:=1000; [seq(f(n),n=1..120)];

NestList[#-Total[IntegerDigits[#]]&,1000,100] (* Harvey P. Dale, Mar 28 2020 *)

A230286 a(n) = A016052(n)/3.

Original entry on oeis.org

1, 2, 4, 5, 7, 8, 10, 11, 13, 17, 19, 23, 28, 32, 37, 38, 40, 41, 43, 47, 49, 53, 58, 62, 67, 68, 70, 71, 73, 77, 79, 83, 88, 92, 97, 101, 103, 107, 109, 113, 118, 122, 127, 131, 136, 140, 142, 146, 151, 155, 160, 164

Offset: 1

Reinhard Zumkeller and N. J. A. Sloane, Oct 16 2013

a(1) = 1; for n >= 1, a(n+1) = a(n) + sum of its digits.

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

Cf. A004207, A016052, A230287 (first differences).

Haskell
```
a230286 = (flip div 3) . a016052
```

cp's OEIS Frontend

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

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A229168 Define a sequence of real numbers by b(1)=2, b(n+1) = b(n) + log_2(b(n)); a(n) = smallest i such that b(i) >= 2^n.

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A219675 Starting with a(0)=0, a(n) = 1 + the sum of the digital sums of a(0) through a(n-1).

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A229171 Define a sequence of real numbers by b(1)=e, b(n+1) = b(n) + log(b(n)); a(n) = smallest i such that b(i) >= e^n.

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A286660 a(n) = a(n-1) + sum of base-100 digits of a(n-1), a(0) = 1.

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A326834 a(0) = 0; a(1) = 0; for n > 0, a(n) = the sum of the number of times each digit in a(n-1) has occurred from a(0) to a(n-2) inclusive.

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A084228 a(1)=1, a(2)=2; thereafter a(n) = sum of digits of (a(1)+a(2)+a(3)+...+a(n-1)).

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