A007618 -id:A007618 - cp's OEIS frontend

A352375 Sum of digits of A007618.

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

Offset: 1

Mateusz Pasternak, Mar 14 2022

D. R. Kaprekar, Puzzles of the Self-Numbers. 311 Devlali Camp, Devlali, India, 1959.

D. R. Kaprekar, The Mathematics of the New Self Numbers, 1963. [annotated and scanned]

Cf. A007618, A007953.

lista(nn) = my(s, x=5); for(n=1, nn, print1(s=sumdigits(x), ", "); x+=s); \\ Jinyuan Wang, Mar 22 2022

from itertools import islice
def A352375_gen(): # generator of terms
    a = 5
    while True:
        yield (s := sum(int(d) for d in str(a)))
        a += s
A352375_list = list(islice(A352375_gen(),20)) # Chai Wah Wu, Mar 29 2022

a(n) = A007953(A007618(n)).

a(n) = A007618(n+1)-A007618(n). - Chai Wah Wu, Mar 29 2022

More terms from Jinyuan Wang, Mar 22 2022

A062028 a(n) = n + sum of the digits of n.

Original entry on oeis.org

0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 77

Offset: 0

Amarnath Murthy, Jun 02 2001

a(n) = A248110(n,A007953(n)). - Reinhard Zumkeller, Oct 01 2014

			a(34) = 34 + 3 + 4 = 41, a(40) = 40 + 4 = 44.

Harry J. Smith and N. J. A. Sloane, Table of n, a(n) for n = 0..10000 (first 1000 terms were computed by Harry J. Smith)
Index entries for Colombian or self numbers and related sequences

Cf. A007953, A006378, A048521, A107740, A066568, A064806, A176995 (range), A003052, A230093, A248110.
Indices of: A047791 (primes), A107743 (composites), A066564 (squares), A084661 (cubes).
Iterations: A004207 (start=1), A016052 (start=3), A007618 (start=5), A006507 (start=7), A016096 (start=9).

a062028 n = a007953 n + n  -- Reinhard Zumkeller, Oct 11 2013

with(numtheory): for n from 1 to 100 do a := convert(n,base,10):
c := add(a[i],i=1..nops(a)): printf(`%d,`,n+c); od:
A062028 := n -> n+add(i,i=convert(n,base,10)) # M. F. Hasler, Nov 08 2018

Table[n + Total[IntegerDigits[n]], {n, 0, 100}]

A062028(n)=n+sumdigits(n) \\ M. F. Hasler, Jul 19 2015

def a(n): return n + sum(map(int, str(n)))
print([a(n) for n in range(71)]) # Michael S. Branicky, Jan 09 2023

a(n) = n + A007953(n).

a(n) = A160939(n+1) - 1. - Filip Zaludek, Oct 26 2016

More terms from Vladeta Jovovic, Jun 05 2001

A004207 a(0) = 1, a(n) = sum of digits of all previous terms.

Original entry on oeis.org

1, 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

Offset: 0

N. J. A. Sloane

If the leading 1 is omitted, this is the important sequence b(1)=1, for n >= 2, b(n) = b(n-1) + sum of digits of b(n-1). Cf. A016052, A016096, etc. - N. J. A. Sloane, Dec 01 2013
Same digital roots as A065075 (Sum of digits of the sum of the preceding numbers) and A001370 (Sum of digits of 2^n); they end in the cycle {1 2 4 8 7 5}. - Alexandre Wajnberg, Dec 11 2005
More precisely, mod 9 this sequence is 1 (1 2 4 8 7 5)*, the parenthesized part being repeated indefinitely. This shows that this sequence is disjoint from A016052. - N. J. A. Sloane, Oct 15 2013
There are infinitely many even terms (Belov 2003).
a(n) = A007618(n-5) for n > 57; a(n) = A006507(n-4) for n > 15. - Reinhard Zumkeller, Oct 14 2013

N. Agronomof, Problem 4421, L'Intermédiaire des mathématiciens, v. 21 (1914), p. 147.
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.
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).
G. E. Stevens and L. G. Hunsberger, A Result and a Conjecture on Digit Sum Sequences, J. Recreational Math. 27, no. 4 (1995), pp. 285-288.
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 37.

T. D. Noe, Table of n, a(n) for n = 0..10000
A. Ya. Belov (ed.), Collection of monster problems in mathematics (in Russian), 2003. Problem 39.
D. R. Kaprekar, The Mathematics of the New Self Numbers [annotated and scanned]
J. Laroche & N. J. A. Sloane, Correspondence, 1977
Project Euler, Problem 551: Sum of digits sequence.
Kenneth B. Stolarsky, The sum of a digitaddition series, Proc. Amer. Math. Soc. 59 (1976), no. 1, 1--5. MR0409340 (53 #13099)
Index entries for Colombian or self numbers and related sequences

Cf. A016052, A016096, A033298, A007612, A007953, A229527, A230107.
For the base-2 analog see A010062.
A065075 gives sum of digits of a(n).
See A219675 for an essentially identical sequence.

a004207 n = a004207_list !! n
a004207_list = 1 : iterate a062028 1
-- Reinhard Zumkeller, Oct 14 2013, Sep 12 2011

read("transforms") :
A004207 := proc(n)
    option remember;
    if n = 0 then
        1;
    else
        add( digsum(procname(i)),i=0..n-1) ;
    end if;
end proc: # R. J. Mathar, Apr 02 2014
# second Maple program:
a:= proc(n) option remember; `if`(n<2, 1, (t->
     t+add(i, i=convert(t, base, 10)))(a(n-1)))
    end:
seq(a(n), n=0..60);  # Alois P. Heinz, Jul 31 2022

f[s_] := Append[s, Plus @@ Flatten[IntegerDigits /@ s]]; Nest[f, {1}, 55] (* Robert G. Wilson v, May 26 2006 *)
f[n_] := n + Plus @@ IntegerDigits@n; Join[{1}, NestList[f, 1, 80]] (* Alonso del Arte, May 27 2006 *)

a(n) = { my(f(d, i) = d+vecsum(digits(d)), S=vector(n)); S[1]=1; for(k=1, n-1, S[k+1] = fold(f, S[1..k])); S } \\ Satish Bysany, Mar 03 2017

a = 1; print1(a, ", "); for(i = 1, 50, print1(a, ", "); a = a + sumdigits(a)); \\ Nile Nepenthe Wynar, Feb 10 2018

from itertools import islice
def agen():
    yield 1; an = 1
    while True: yield an; an += sum(map(int, str(an)))
print(list(islice(agen(), 54))) # Michael S. Branicky, Jul 31 2022

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

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

Errors from 25th term on corrected by Leonid Broukhis, Mar 15 1996

Typo in definition fixed by Reinhard Zumkeller, Sep 14 2011

A016052 a(1) = 3; for n >= 1, a(n+1) = a(n) + sum of its digits.

Original entry on oeis.org

3, 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

Offset: 1

Robert G. Wilson v

Mod 9 this sequence is 3, 6, 3, 6, 3, 6, ... This shows that this sequence is disjoint from A004207. - N. J. A. Sloane, Oct 15 2013

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.
G. E. Stevens and L. G. Hunsberger, A Result and a Conjecture on Digit Sum Sequences, J. Recreational Math. 27, no. 4 (1995), pp. 285-288.
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, pages 34-35.

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, A007618, A006507, A016096, A230107, A230286, A230287, A084228.

a016052 n = a016052_list !! (n-1)
a016052_list = iterate a062028 3  -- Reinhard Zumkeller, Oct 14 2013

NestList[# + Total[IntegerDigits[#]] &, 3, 51] (* Jayanta Basu, Aug 11 2013 *)
a[1] = 3; a[n_] := a[n] = a[n - 1] + Total@ IntegerDigits@ a[n - 1]; Array[a, 80] (* Robert G. Wilson v, Jun 27 2014 *)

a_list(nn) = { my(f(n, i) = n + vecsum(digits(n)), S=vector(nn+1)); S[1]=3; for(k=2, #S, S[k] = fold(f, S[1..k-1])); S[2..#S] } \\ Satish Bysany, Mar 04 2017

from itertools import islice
def A016052_gen(): # generator of terms
    yield (a:=3)
    while True: yield (a:=a+sum(map(int,str(a))))
A016052_list = list(islice(A016052_gen(),20)) # Chai Wah Wu, Jun 16 2024

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

a(n) - a(n-1) = A084228(n+1). - Robert G. Wilson v, Jun 27 2014

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

Original entry on oeis.org

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

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

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 *)

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.

Original entry on oeis.org

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

A129888 Start with 10; write down the sum of its digits; add last two terms; repeat.

Original entry on oeis.org

10, 1, 11, 2, 13, 4, 17, 8, 25, 7, 32, 5, 37, 10, 47, 11, 58, 13, 71, 8, 79, 16, 95, 14, 109, 10, 119, 11, 130, 4, 134, 8, 142, 7, 149, 14, 163, 10, 173, 11, 184, 13, 197, 17, 214, 7, 221, 5, 226, 10, 236, 11, 247, 13, 260, 8, 268, 16, 284, 14, 298, 19, 317, 11, 328, 13

Offset: 1

N. J. A. Sloane, May 26 2007

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Cf. A007618, A065075.

nxt[n_]:=Module[{t=Total[n]},{t,Total[IntegerDigits[t]]}]; Flatten[ NestList[ nxt,{10,1},35]] (* Harvey P. Dale, Mar 29 2011 *)

def next2(n): sd = sum(map(int, str(n))); return [sd, n+sd]
def aupton(terms):
    alst = [10]
    while len(alst) < terms: alst.extend(next2(alst[-1]))
    return alst[:terms]
print(aupton(66)) # Michael S. Branicky, Oct 01 2021

a(2n+1)=A007618(n+2). For 1<=n<=10: a(2n)=A065075(n+1). - R. J. Mathar, Jun 14 2007

More terms from R. J. Mathar, Jun 14 2007

cp's OEIS Frontend

A352375 Sum of digits of A007618.

Views

Author

Keywords

References

Links

Crossrefs

Programs

PARI

Python

Formula

Extensions

A062028 a(n) = n + sum of the digits of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

Python

Formula

Extensions

A004207 a(0) = 1, a(n) = sum of digits of all previous terms.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

PARI

Python

Formula

Extensions

A016052 a(1) = 3; for n >= 1, a(n+1) = a(n) + sum of its digits.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

Python

Formula

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

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Haskell

Mathematica

Formula

Extensions

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

Views

Author

Keywords

References

Links

Crossrefs