A016052 -id:A016052 - cp's OEIS frontend

A230107 Define a sequence by b(1)=n, b(k+1)=b(k)+(sum of digits of b(k)); a(n) is the number of steps needed to reach a term in A004207, or a(n) = -1 if the sequence never joins A004207.

0, 0, -1, 0, 52, -1, 11, 0, -1, 51, 50, -1, 49, 10, -1, 0, 48, -1, 9, 50, -1, 49, 0, -1, 47, 48, -1, 0, 8, -1, 49, 46, -1, 47, 48, -1, 45, 0, -1, 7, 46, -1, 47, 6, -1, 45, 44, -1, 0, 46, -1, 5, 5, -1, 45, 44, -1, 43, 4, -1, 4, 0, -1, 4, 44, -1, 43, 3, -1, 0

Offset: 0

N. J. A. Sloane and Reinhard Zumkeller, Oct 15 2013; corrected Oct 20 2013

Looking at b(k) mod 9 shows that a(n) = -1 whenever n is a multiple of 3 (since then the b sequence is disjoint from A004207).

Conjecture: the b sequence, for any starting value n, will eventually merge with one of A000004 (the zero sequence), A004207, A016052 or A016096.

			For n=3, A016052 never meets A004207, so a(3) = -1.
For n=5, A007618 meets A004207 at the 53rd term, 620, so a(5) = 53.

Index entries for Colombian or self numbers and related sequences

Cf. A004207, A016052, A007618, A006507, A016096, A062028, A230299.

import Data.Maybe (fromMaybe)
a230107 = fromMaybe (-1) . f (10^5) 1 1 1 where
   f k i u j v | k <= 0    = Nothing
               | u < v     = f (k - 1) (i + 1) (a062028 u) j v
               | u > v     = f (k - 1) i u (j + 1) (a062028 v)
               | otherwise = Just j

read transforms; # to get digsum
M:=2000;
# f(s) returns the sequence k->k+digsum(k) starting at s
f:=proc(s) global M; option remember; local n,k,s1;
s1:=[s]; k:=s;
for n from 1 to M do  k:=k+digsum(k);
s1:=[op(s1),k]; od: end;
# g(s) returns (x,p), where x = first number in common between
# f(1) and f(s), and p is the position where it occurred.
# If f(1), f(s) are disjoint for M terms, returns (-1,-1)
S1:=convert(f(1),set):
g:=proc(s) global f,S1; local t1,p,S2,S3;
S2:=convert(f(s),set);
S3:= S1 intersect S2;
t1:=min(S3);
if (t1 = infinity) then RETURN(-1,-1); else
  member(t1,f(s),'p'); RETURN(t1,p-1); fi;
end;
[seq(g(n)[2],n=1..20)];

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

A140131 a(n) = a(n-1) + a(n-2) + digsum(a(n-1)) + digsum(a(n-2)), with a(0)=0 and a(1)=1.

Original entry on oeis.org

0, 1, 2, 6, 16, 35, 66, 121, 203, 333, 550, 902, 1473, 2401, 3896, 6330, 10264, 16619, 26919, 43588, 70562, 114198, 184804, 299051, 483906, 783013, 1266971, 2050038, 3317059, 5367143, 8684259, 14051473, 22735799, 36787341, 59523223, 96310634, 155833920, 252144622

Offset: 0

Paolo P. Lava and Giorgio Balzarotti, May 09 2008

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Cf. A007953, A016052, A047892, A047897-A047900, A047902, A047903, A055263, A134268, A135210, A140132.

P:=proc(n) local i,t; t:=[0,1]; for i from 1 to n do t:=[op(t),t[-2]+t[-1]+convert(convert(t[-2],base,10),`+`)+convert(convert(t[-1],base,10),`+`)]; od; print(op(t)); end: P(34); # Paolo P. Lava, Jun 25 2024

nxt[{a_,b_}]:=a+b+Total[IntegerDigits[a]]+Total[IntegerDigits[b]]; Transpose[NestList[{Last[#],nxt[#]}&,{0,1},40]][[1]] (* Harvey P. Dale, Oct 31 2011 *)

A230299 Define a sequence b_s by b_s(1)=s, b_s(k+1)=b_s(k)+(sum of digits of b_s(k)); a(n) is the number of steps needed for b_n to reach a term in one of b_0, b_1, b_3 or b_9, or a(n) = -1 if b_n never joins one of these four sequences.

Original entry on oeis.org

0, 0, 0, 0, 0, 52, 0, 11, 0, 0, 51, 50, 0, 49, 10, 0, 0, 48, 0, 9, 50, 0, 49, 0, 0, 47, 48, 0, 0, 8, 0, 49, 46, 0, 47, 48, 0, 45, 0, 0, 7, 46, 7, 47, 6, 0, 45, 44, 6, 0, 46, 0, 5, 5, 0, 45, 44, 0, 43, 4, 5, 4, 0, 0, 4, 44, 4, 43, 3, 0, 0, 42, 0, 3, 3, 4, 43, 0

Offset: 0

N. J. A. Sloane and Reinhard Zumkeller, Oct 21 2013

We conjecture that a(n) is never -1.

Cf. A230107, A062028, A004207, A016052, A007618, A006507, A016096.

read transforms; # to get digsum
M:=2000;
# f(s) returns the sequence k->k+digsum(k) starting at s
f:=proc(s) global M; option remember; local n,k,s1;
s1:=[s]; k:=s;
for n from 1 to M do  k:=k+digsum(k);
s1:=[op(s1),k]; od: end;
# g(s) returns (x,p), where x = first number in common between
# f(s) and one of f(1), f(3), f(9) and p is the position where it occurred.
# If f(s) and all of f(1), f(3), f(9) are disjoint for M terms, returns (-1,-1)
S1:=convert(f(1),set):
S3:=convert(f(3),set):
S9:=convert(f(9),set):
g:=proc(s) global f,S1,S3,S9; local t1,p,T0,T1,T2;
T0:=f(s):
T1:=convert(T0,set);
if ((s mod 9) = 3) or ((s mod 9) = 6) then   T2:= T1 intersect S3;   t1:=min(T2);   if (t1 = infinity) then RETURN(-1,-1); else     member(t1,T0,'p'); RETURN(t1,p-1); fi;
elif ((s mod 9) = 0) then   T2:= T1 intersect S9;   t1:=min(T2);   if (t1 = infinity) then RETURN(-1,-1); else     member(t1,T0,'p'); RETURN(t1,p-1); fi;
else   T2:= T1 intersect S1;   t1:=min(T2);   if (t1 = infinity) then RETURN(-1,-1); else     member(t1,T0,'p'); RETURN(t1,p-1); fi;
fi;
end;
[seq(g(n)[2],n=1..45)];

Terms a(46) and beyond from Lars Blomberg, Jan 10 2018

A249043 a(1) = 42; a(n+1) = a(n) + sum of decimal digits of a(n).

Original entry on oeis.org

42, 48, 60, 66, 78, 93, 105, 111, 114, 120, 123, 129, 141, 147, 159, 174, 186, 201, 204, 210, 213, 219, 231, 237, 249, 264, 276, 291, 303, 309, 321, 327, 339, 354, 366, 381, 393, 408, 420, 426, 438, 453, 465, 480, 492, 507, 519, 534, 546, 561, 573, 588, 609, 624, 636

Offset: 1

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.

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, A016052, A036228.

Cf. A007953, A062028.

a249043 n = a249043_list !! (n-1)
a249043_list = iterate a062028 42
-- Reinhard Zumkeller, Oct 31 2014

a(n+1) = A062028(a(n)). - Reinhard Zumkeller, Oct 31 2014

A036227 a(1) = 20; a(n+1) = a(n) + sum of decimal digits of a(n).

Original entry on oeis.org

20, 22, 26, 34, 41, 46, 56, 67, 80, 88, 104, 109, 119, 130, 134, 142, 149, 163, 173, 184, 197, 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 >= 109 can be found in A007618.

Index entries for Colombian or self numbers and related sequences

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

NestList[#+Total[IntegerDigits[#]]&,20,60] (* Harvey P. Dale, May 11 2014 *)

Edited by Charles R Greathouse IV, Aug 02 2010

A036233 Inverse Colombian function.

Original entry on oeis.org

1, 1, 3, 1, 5, 3, 7, 1, 9, 5, 5, 3, 5, 7, 3, 1, 5, 9, 7, 20, 3, 20, 1, 3, 5, 20, 9, 1, 7, 3, 31, 5, 3, 20, 31, 9, 5, 1, 3, 7, 20, 42, 31, 7, 9, 20, 5, 42, 1, 31, 3, 7, 53, 9, 31, 20, 3, 5, 7, 42, 53, 1, 9, 64, 31, 42, 20, 53, 3, 1, 5, 9, 7, 64, 75, 31, 1, 42, 5, 20, 9, 53, 7, 3, 64, 86, 75, 20

Offset: 1

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

a(n) is the smallest x with n in the digit summing sequence starting with x.

Contains only self-numbers, see A003052.

Sean A. Irvine, Table of n, a(n) for n = 1..10000
Sean A. Irvine, Java program (github)

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

A123171 a(1) = 123, a(n) = sum of digits of all previous terms.

Original entry on oeis.org

123, 6, 12, 15, 21, 24, 30, 33, 39, 51, 57, 69, 84, 96, 111, 114, 120, 123, 129, 141, 147, 159, 174, 186, 201, 204, 210, 213, 219, 231, 237, 249, 264, 276, 291, 303, 309, 321, 327, 339, 354, 366, 381, 393, 408, 420, 426, 438, 453, 465, 480, 492, 507, 519, 534

Offset: 1

Herman Jamke (hermanjamke(AT)fastmail.fm), Oct 02 2006

a(1) = 123 a(2) = 1 + 2 + 3 = 6 a(3) = (1 + 2 + 3) + 6 = 12 a(4) = (1 + 2 + 3) + 6 + (1 + 2) = 15 a(5) = (1 + 2 + 3) + 6 + (1 + 2) + (1 + 5) = 21 a(6) = (1 + 2 + 3) + 6 + (1 + 2) + (1 + 5) + (2 + 1) = 24 a(7) = (1 + 2 + 3) + 6 + (1 + 2) + (1 + 5) + (2 + 1) + (2 + 4) = 30 ...

Essentially the same as A016052. - R. J. Mathar, Jun 18 2008

James C. McMahon, Table of n, a(n) for n = 1..10000
Sergio Silva, Teste Numerico.

Cf. A004207.

s={123};sum=0;Do[sum=sum+Total[IntegerDigits[s[[-1]]]];AppendTo[s,sum],{n,54}];s (* James C. McMahon, Nov 16 2024 *)

s=0; a=123; print1(a,","); for(n=1,100,dig=eval(Vec(Str(a)));s=s+sum(i=1,length(dig),dig[i]);print1(s,",");a=s)

a(n) = a(n-1) + sum of digits of a(n-1), a(1) = 123

A140132 a(n) = Sum_digits{a(n-1)+a(n-2)+Sum_digits[a(n-1)]+Sum_digits[a(n-2)]}, with a(0)=0 and a(1)=1.

Original entry on oeis.org

0, 1, 2, 6, 7, 8, 3, 4, 5, 9, 10, 11, 6, 7, 8, 3, 4, 5, 9, 10, 11, 6, 7, 8, 3, 4, 5, 9, 10, 11, 6, 7, 8, 3, 4, 5, 9, 10, 11, 6, 7, 8, 3, 4, 5, 9, 10, 11, 6, 7, 8, 3, 4, 5, 9, 10, 11, 6, 7, 8, 3, 4, 5, 9, 10, 11, 6, 7, 8, 3, 4, 5, 9, 10, 11, 6, 7, 8, 3, 4, 5, 9, 10, 11, 6, 7, 8, 3, 4, 5, 9, 10, 11, 6, 7

Offset: 0

Paolo P. Lava and Giorgio Balzarotti, May 09 2008

After the first three terms the sequence is periodic: 6,7,8,3,4,5,9,10,11.

Cf. A016052, A047892, A047897, A047898, A047899, A047900, A047902, A047903, A055263, A134268, A135210, A140131.

P:=proc(n) local a,i,t; t:=[0,1]; for i from 1 to n do
a:=t[-2]+t[-1]+convert(convert(t[-2],base,10),`+`)+convert(convert(t[-1],base,10),`+`);
t:=[op(t),convert(convert(a,base,10),`+`)]; od; print(op(t)); end: P(93); # Paolo P. Lava, Jun 25 2024

A151942 Table of Self / Colombian numbers and their descendents.

Original entry on oeis.org

1, 2, 3, 4, 6, 5, 8, 12, 10, 7, 16, 15, 11, 14, 9, 23, 21, 13, 19, 18, 20, 28, 24, 17, 29, 27, 22, 31, 38, 30, 25, 40, 36, 26, 35, 42, 49, 33, 32, 44, 45, 34, 43, 48, 53, 62, 39, 37, 52, 54, 41, 50, 60, 61, 64, 70, 51, 47, 59, 63, 46, 55, 66, 68, 74, 75, 77, 57, 58, 73, 72, 56

Offset: 1

Carl R. White, Jul 13 2009

Initially resembles a permutation of the integers, but this is not the case. 101 is the first number to appear twice, descending from both 91 and 100: 91 + 9+1 = 100 + 1+0+0 = 101

D. R. Kaprekar, The Mathematics of the New Self Numbers [annotated and scanned]
Index entries for Colombian or self numbers and related sequences

First column of table is the Self numbers: A003052; First through eighth rows are A004207, A016052, A007618, A006507, A016096, A036227, A036228 respectively.

T(r,0) are those numbers not of form n + sum of digits of n (Self numbers)

T(r,c) = T(r,c-1) + sum of digits of T(r,c-1)

cp's OEIS Frontend

A230107 Define a sequence by b(1)=n, b(k+1)=b(k)+(sum of digits of b(k)); a(n) is the number of steps needed to reach a term in A004207, or a(n) = -1 if the sequence never joins A004207.

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

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A140131 a(n) = a(n-1) + a(n-2) + digsum(a(n-1)) + digsum(a(n-2)), with a(0)=0 and a(1)=1.

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A230299 Define a sequence b_s by b_s(1)=s, b_s(k+1)=b_s(k)+(sum of digits of b_s(k)); a(n) is the number of steps needed for b_n to reach a term in one of b_0, b_1, b_3 or b_9, or a(n) = -1 if b_n never joins one of these four sequences.

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A249043 a(1) = 42; a(n+1) = a(n) + sum of decimal digits of a(n).

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A036227 a(1) = 20; a(n+1) = a(n) + sum of decimal digits of a(n).

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A036233 Inverse Colombian function.

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A123171 a(1) = 123, a(n) = sum of digits of all previous terms.

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