A062028 -id:A062028 - cp's OEIS frontend

A007953 Digital sum (i.e., sum of digits) of n; also called digsum(n).

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 8, 9, 10, 11, 12, 13, 14, 15

Offset: 0

R. Muller

Do not confuse with the digital root of n, A010888 (first term that differs is a(19)).
Also the fixed point of the morphism 0 -> {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}, 1 -> {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, 2 -> {2, 3, 4, 5, 6, 7, 8, 9, 10, 11}, etc. - Robert G. Wilson v, Jul 27 2006
For n < 100 equal to (floor(n/10) + n mod 10) = A076314(n). - Hieronymus Fischer, Jun 17 2007
It appears that a(n) is the position of 10*n in the ordered set of numbers obtained by inserting/placing one digit anywhere in the digits of n (except a zero before 1st digit). For instance, for n=2, the resulting set is (12, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 32, 42, 52, 62, 72, 82, 92) where 20 is at position 2, so a(2) = 2. - Michel Marcus, Aug 01 2022
Also the total number of beads required to represent n on a Russian abacus (schoty). - P. Christopher Staecker, Mar 31 2023
a(n) / a(2n) <= 5 with equality iff n is in A169964, while a(n) / a(3n) is unbounded, since if n = (10^k + 2)/3, then a(n) = 3*k+1, a(3n) = 3, so a(n) / a(3n) = k + 1/3 -> oo when k->oo (see Diophante link). - Bernard Schott, Apr 29 2023
Also the number of symbols needed to write number n in Egyptian numerals for n < 10^7. - Wojciech Graj, Jul 10 2025

			a(123) = 1 + 2 + 3 = 6, a(9875) = 9 + 8 + 7 + 5 = 29.

N. J. A. Sloane, Table of n, a(n) for n = 0..10000
Krassimir Atanassov, On the 16th Smarandache Problem, Notes on Number Theory and Discrete Mathematics, Sophia, Bulgaria, Vol. 5 (1999), No. 1, 36-38.
Krassimir Atanassov, On Some of the Smarandache's Problems, 1999.
Jean-Luc Baril, Classical sequences revisited with permutations avoiding dotted pattern, Electronic Journal of Combinatorics, Vol. 18 (2011), #P178.
F. M. Dekking, The Thue-Morse Sequence in Base 3/2, J. Int. Seq., Vol. 26 (2023), Article 23.2.3.
Diophante, A1762, Des chiffres à la moulinette (in French).
Ernesto Estrada and Puri Pereira-Ramos, Spatial 'Artistic' Networks: From Deconstructing Integer-Functions to Visual Arts, Complexity, Vol. 2018 (2018), Article ID 9893867.
A. O. Gel'fond, Sur les nombres qui ont des propriétés additives et multiplicatives données (French) Acta Arith., Vol. 13 (1967/1968), pp. 259-265. MR0220693 (36 #3745)
Christian Mauduit and András Sárközy, On the arithmetic structure of sets characterized by sum of digits properties J. Number Theory, Vol. 61, No. 1 (1996), pp. 25-38. MR1418316 (97g:11107)
Christian Mauduit and András Sárközy, On the arithmetic structure of the integers whose sum of digits is fixed, Acta Arith., Vol. 81, No. 2 (1997), pp. 145-173. MR1456239 (99a:11096)
Kerry Mitchell, Spirolateral-Type Images from Integer Sequences, 2013.
Kerry Mitchell, Spirolateral image for this sequence . [taken, with permission, from the Spirolateral-Type Images from Integer Sequences article]
Jan-Christoph Puchta and Jürgen Spilker, Altes und Neues zur Quersumme, Mathematische Semesterberichte, Vol. 49 (2002), pp. 209-226.
Jan-Christoph Puchta and Jürgen Spilker, Altes und Neues zur Quersumme.
Maxwell Schneider and Robert Schneider, Digit sums and generating functions, arXiv:1807.06710 [math.NT], 2018.
Jeffrey O. Shallit, Problem 6450, Advanced Problems, The American Mathematical Monthly, Vol. 91, No. 1 (1984), pp. 59-60; Two series, solution to Problem 6450, ibid., Vol. 92, No. 7 (1985), pp. 513-514.
Vladimir Shevelev, Compact integers and factorials, Acta Arith., Vol. 126, No. 3 (2007), pp. 195-236 (cf. pp. 205-206).
Robert Walker, Self Similar Sloth Canon Number Sequences.
Eric Weisstein's World of Mathematics, Digit Sum.
Wikipedia, Digit sum.
Index entries for Colombian or self numbers and related sequences

Cf. A003132, A055012, A055013, A055014, A055015, A010888, A007954, A031347, A055017, A076313, A076314, A054899, A138470, A138471, A138472, A000120, A004426, A004427, A054683, A054684, A069877, A179082-A179085, A108971, A169964, A179987, A179988, A180018, A180019, A217928, A216407, A037123, A074784, A231688, A231689, A225693, A254524 (ordinal transform).
Bisections: A004092, A004155.
For n + digsum(n) see A062028.

a007953 n | n < 10 = n
          | otherwise = a007953 n' + r where (n',r) = divMod n 10
-- Reinhard Zumkeller, Nov 04 2011, Mar 19 2011

[ &+Intseq(n): n in [0..87] ];  // Bruno Berselli, May 26 2011

A007953 := proc(n) add(d,d=convert(n,base,10)) ; end proc: # R. J. Mathar, Mar 17 2011

Table[Sum[DigitCount[n][[i]] * i, {i, 9}], {n, 50}] (* Stefan Steinerberger, Mar 24 2006 *)
Table[Plus @@ IntegerDigits @ n, {n, 0, 87}] (* or *)
Nest[Flatten[# /. a_Integer -> Array[a + # &, 10, 0]] &, {0}, 2] (* Robert G. Wilson v, Jul 27 2006 *)
Total/@IntegerDigits[Range[0,90]] (* Harvey P. Dale, May 10 2016 *)
DigitSum[Range[0, 100]] (* Requires v. 14 *) (* Paolo Xausa, May 17 2024 *)

a(n)=if(n<1, 0, if(n%10, a(n-1)+1, a(n/10))) \\ Recursive, very inefficient. A more efficient recursive variant: a(n)=if(n>9, n=divrem(n, 10); n[2]+a(n[1]), n)

a(n, b=10)={my(s=(n=divrem(n, b))[2]); while(n[1]>=b, s+=(n=divrem(n[1], b))[2]); s+n[1]} \\ M. F. Hasler, Mar 22 2011

a(n)=sum(i=1, #n=digits(n), n[i]) \\ Twice as fast. Not so nice but faster:

a(n)=sum(i=1,#n=Vecsmall(Str(n)),n[i])-48*#n \\ M. F. Hasler, May 10 2015
/* Since PARI 2.7, one can also use: a(n)=vecsum(digits(n)), or better: A007953=sumdigits. [Edited and commented by M. F. Hasler, Nov 09 2018] */

a(n) = sumdigits(n); \\ Altug Alkan, Apr 19 2018

def A007953(n):
    return sum(int(d) for d in str(n)) # Chai Wah Wu, Sep 03 2014

def a(n): return sum(map(int, str(n))) # Michael S. Branicky, May 22 2021

(0 to 99).map(.toString.map(.toInt - 48).sum) // Alonso del Arte, Sep 15 2019

"Recursive version for general bases. Set base = 10 for this sequence."
digitalSum: base
| s |
base = 1 ifTrue: [^self].
(s := self // base) > 0
  ifTrue: [^(s digitalSum: base) + self - (s * base)]
  ifFalse: [^self]
"by Hieronymus Fischer, Mar 24 2014"

A007953(n): String(n).compactMap{$0.wholeNumberValue}.reduce(0, +) // Egor Khmara, Jun 15 2021

a(A051885(n)) = n.
a(n) <= 9(log_10(n)+1). - Stefan Steinerberger, Mar 24 2006
From Benoit Cloitre, Dec 19 2002: (Start)
a(0) = 0, a(10n+i) = a(n) + i for 0 <= i <= 9.
a(n) = n - 9*(Sum_{k > 0} floor(n/10^k)) = n - 9*A054899(n). (End)
From Hieronymus Fischer, Jun 17 2007: (Start)
G.f. g(x) = Sum_{k > 0, (x^k - x^(k+10^k) - 9x^(10^k))/(1-x^(10^k))}/(1-x).
a(n) = n - 9*Sum_{10 <= k <= n} Sum_{j|k, j >= 10} floor(log_10(j)) - floor(log_10(j-1)). (End)
From Hieronymus Fischer, Jun 25 2007: (Start)
The g.f. can be expressed in terms of a Lambert series, in that g(x) = (x/(1-x) - 9*L[b(k)](x))/(1-x) where L[b(k)](x) = sum{k >= 0, b(k)*x^k/(1-x^k)} is a Lambert series with b(k) = 1, if k > 1 is a power of 10, else b(k) = 0.
G.f.: g(x) = (Sum_{k > 0} (1 - 9*c(k))*x^k)/(1-x), where c(k) = Sum_{j > 1, j|k} floor(log_10(j)) - floor(log_10(j-1)).
a(n) = n - 9*Sum_{0 < k <= floor(log_10(n))} a(floor(n/10^k))*10^(k-1). (End)
From Hieronymus Fischer, Oct 06 2007: (Start)
a(n) <= 9*(1 + floor(log_10(n))), equality holds for n = 10^m - 1, m > 0.
lim sup (a(n) - 9*log_10(n)) = 0 for n -> oo.
lim inf (a(n+1) - a(n) + 9*log_10(n)) = 1 for n -> oo. (End)
a(n) = A138530(n, 10) for n > 9. - Reinhard Zumkeller, Mar 26 2008
a(A058369(n)) = A004159(A058369(n)); a(A000290(n)) = A004159(n). - Reinhard Zumkeller, Apr 25 2009
a(n) mod 2 = A179081(n). - Reinhard Zumkeller, Jun 28 2010
a(n) <= 9*log_10(n+1). - Vladimir Shevelev, Jun 01 2011
a(n) = a(n-1) + a(n-10) - a(n-11), for n < 100. - Alexander R. Povolotsky, Oct 09 2011
a(n) = Sum_{k >= 0} A031298(n, k). - Philippe Deléham, Oct 21 2011
a(n) = a(n mod b^k) + a(floor(n/b^k)), for all k >= 0. - Hieronymus Fischer, Mar 24 2014
Sum_{n>=1} a(n)/(n*(n+1)) = 10*log(10)/9 (Shallit, 1984). - Amiram Eldar, Jun 03 2021

More terms from Hieronymus Fischer, Jun 17 2007

Edited by Michel Marcus, Nov 11 2013

A003052 Self numbers or Colombian numbers (numbers that are not of the form m + sum of digits of m for any m).

Original entry on oeis.org

1, 3, 5, 7, 9, 20, 31, 42, 53, 64, 75, 86, 97, 108, 110, 121, 132, 143, 154, 165, 176, 187, 198, 209, 211, 222, 233, 244, 255, 266, 277, 288, 299, 310, 312, 323, 334, 345, 356, 367, 378, 389, 400, 411, 413, 424, 435, 446, 457, 468, 479, 490, 501, 512, 514, 525

Offset: 1

N. J. A. Sloane

From Amiram Eldar, Nov 28 2020: (Start)
The term "self numbers" was coined by Kaprekar (1959). The term "Colombian number" was coined by Recamán (1973) of Bogota, Colombia.
The asymptotic density of this sequence is approximately 0.0977778 (Guaraldo, 1978). (End)

Steven R. Finch, Mathematical Constants, Cambridge, 2003, Section 2.24.
Martin Gardner, Time Travel and Other Mathematical Bewilderments. Freeman, NY, 1988, p. 116.
V. S. Joshi, A note on self-numbers. Volume dedicated to the memory of V. Ramaswami Aiyar. Math. Student, Vol. 39 (1971), pp. 327-328. 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.
D. R. Kaprekar, The Mathematics of the New Self Numbers (Part V). 311 Devlali Camp, Devlali, India, 1967.
Bernardo Recamán, The Bogota Puzzles, Dover Publications, Inc., 2020, chapter 36, p. 33.
József Sándor and Borislav Crstici, Handbook of Number theory II, Kluwer Academic Publishers, 2004, Chapter 4, pp. 384-386.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Author?, J. Recreational Math., vol. 23, no. 1, p. 244, 1991.

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.
Christian N. K. Anderson, Ulam Spiral of the first 5000 self numbers.
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]
Martin Gardner and N. J. A. Sloane, Correspondence, 1973-74
Rosalind Guaraldo, On the Density of the Image Sets of Certain Arithmetic Functions - II, The Fibonacci Quarterly, Vol. 16, No. 5 (1978), pp. 481-488.
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, 1963. [annotated and scanned]
Andrzej Makowski, On Kaprekar's "junction numbers", Math. Student, Vol. 34, No. 2 (1966), p. 77. MR0223292 (36 #6340); entire issue.
A. Narasinga Rao, On a technique for obtaining numbers with a multiplicity of generators, Math. Student, Vol. 34, No. 2 (1966), pp. 79-84. MR0229573 (37 #5147); entire issue.
Bernardo Recamán, Problem E2408, Amer. Math. Monthly, Vol. 80, No. 4 (1973), p. 434; Colombian Numbers, solution to Problem E2408 by D. W. Bange, ibid., Vol. 81, No. 4 (1974), p. 407.
Giovanni Resta, Self or Colombian numbers, Numbersaplenty, 2013.
Richard Schorn, Kaprekar's Sequence and his "Selfnumbers", DERIVE Newsletter, #53 (2004), pp. 30-32.
Walter Schneider, Self Numbers, 2000-2003.
Walter Schneider, Self Numbers, 2000-2003 (unpublished; local copy)
N. J. A. Sloane, Martin Gardner and D. R. Kaprekar, Correspondence, 1974 [Scanned letters]
Terry Trotter, Charlene Numbers
Eric Weisstein's World of Mathematics, Self Number.
Wikipedia, Self number.
U. Zannier, On the distribution of self-numbers, Proc. Amer. Math. Soc., Vol. 85, No. 1 (1982), pp. 10-14.
Index entries for Colombian or self numbers and related sequences

Cf. A006886, A232229, A062028, A055642, A282711.
For self primes, i.e., self numbers which are primes, see A006378.
Complement of A176995.
See A010061 for the binary version, A283002 for a base-100 version.
Cf. A247104 (subsequence of squarefree terms).
Cf. A377472 for first differences, A377474 for indices where new differences appear.

a003052 n = a003052_list !! (n-1)
a003052_list = filter ((== 0) . a230093) [1..]
-- Reinhard Zumkeller, Oct 11 2013, Aug 21 2011

isA003052 := proc(n) local k ; for k from 0 to n do if k+A007953(k) = n then RETURN(false): fi; od: RETURN(true) ; end:
A003052 := proc(n) option remember; if n = 1 then 1; else for a from procname(n-1)+1 do if isA003052(a) then RETURN(a) ; fi; od; fi; end: # R. J. Mathar, Jul 27 2009

nn = 525; Complement[Range[nn], Union[Table[n + Total[IntegerDigits[n]], {n, nn}]]] (* T. D. Noe, Mar 31 2013 *)

is_A003052(n)={for(i=1,min(n\2,9*#digits(n)), sumdigits(n-i)==i && return); n}  \\ M. F. Hasler, Mar 20 2011, updated Nov 08 2018

is(n) = {if(n < 30, return((n < 10 && n%2 == 1) || n == 20)); qd = 1 + logint(n, 10); r = 1 + (n-1)%9; h = (r + 9 * (r%2))/2; ld = 10; while(h + 9*qd >= n % ld, ld*=10); vs = vecsum(digits(n \ ld)); n %= ld; for(i = 0, qd, if(vs + vecsum(digits(n - h - 9*i)) == h + 9*i, return(0))); 1} \\ David A. Corneth, Aug 20 2020

A230093(a(n)) = 0. - Reinhard Zumkeller, Oct 11 2013

In fact this defines the sequence: x is in the sequence iff A230093(x) = 0. - M. F. Hasler, Nov 08 2018

More terms from James Sellers, Jul 06 2000

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

A092391 a(n) = n + wt(n), where wt(n) = A000120(n) = binary weight of n.

Original entry on oeis.org

0, 2, 3, 5, 5, 7, 8, 10, 9, 11, 12, 14, 14, 16, 17, 19, 17, 19, 20, 22, 22, 24, 25, 27, 26, 28, 29, 31, 31, 33, 34, 36, 33, 35, 36, 38, 38, 40, 41, 43, 42, 44, 45, 47, 47, 49, 50, 52, 50, 52, 53, 55, 55, 57, 58, 60, 59, 61, 62, 64, 64, 66, 67, 69, 65, 67, 68, 70, 70, 72, 73, 75

Offset: 0

Reinhard Zumkeller, May 08 2004

Reinhard Zumkeller and Donovan Johnson, Table of n, a(n) for n = 0..10000 (terms up to a(1023) from Reinhard Zumkeller)
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.
Index entries for Colombian or self numbers and related sequences

A010061 gives the numbers not occurring in this sequence. A228082 gives the terms of this sequence sorted into ascending order, with duplicates removed. A228085(n) gives the number of times n occurs in this sequence.
Cf. A000120, A228083, A228086, A228087, A228091, A011371, A230058, A230092, A062028.
Cf. A228088, A230091, A227915, A230300.

a092391 n = n + a000120 n  -- Reinhard Zumkeller, May 13 2012

Table[n + Total[IntegerDigits[n, 2]], {n, 0, 100}] (* Jean-François Alcover, Sep 03 2013 *)

A092391(n)=n+hammingweight(n) \\ M. F. Hasler, Oct 05 2013

def a(n): return n + bin(n).count("1")
print([a(n) for n in range(72)]) # Michael S. Branicky, May 26 2022

a(n) = n + A000120(n).
A010062(n+1) = a(A010062(n)).
G.f.: (1/(1 - x))*Sum_{k>=0} (2^k + 1)*x^(2^k)/(1 + x^(2^k)). - Ilya Gutkovskiy, Jul 23 2017

A048519 Prime plus its digit sum equals a prime.

Original entry on oeis.org

11, 13, 19, 37, 53, 59, 71, 73, 97, 101, 103, 127, 149, 163, 167, 181, 233, 257, 271, 277, 293, 307, 367, 383, 389, 419, 431, 433, 479, 499, 509, 547, 563, 587, 617, 631, 701, 727, 743, 787, 811, 839, 857, 859, 947, 1009, 1049, 1061, 1087, 1153, 1171

Offset: 1

Patrick De Geest, May 15 1999

For any prime p, p +- digitsum(p, base b) can't be prime unless the base b is even, since in an odd base, an odd number always has an odd digit sum (powers of b are congruent to b (mod 2)), so p +- digitsum(p, base b) is even for odd b. This sequence is for b = 10 (where "-" is also excluded, see comment in A243442), see A243441 for b = 2. - M. F. Hasler, Nov 06 2018

See subsequence A048523 for primes which only once give another prime under iteration of A062028, and A048524 .. A048527, A320878 .. A320880 for primes starting longer chains. See A090009 for their initial terms, starting the earliest chain of given length. - M. F. Hasler, Nov 09 2018

			a(9) = prime 97 because 97 + sum-of-digits(97) = 97 + 16 = 113 also a prime.

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

Cf. A007953 (digit sum), A062028 (n + digit sum of n), A047791 (A062028(n) is prime), A048520.
Cf. A006378, A107740.
Cf. A000040, A010051, A065073.
Cf. A048523 .. A048527, A320878, A320879, A320880, A090009.

a048519 n = a048519_list !! (n-1)
a048519_list = map a000040 $ filter ((== 1) . a010051' . a065073) [1..]
-- Reinhard Zumkeller, Sep 27 2014

[p: p in PrimesUpTo(1200) | IsPrime(q) where q is p+&+Intseq(p)]; // Vincenzo Librandi, Jan 30 2018

select(n -> isprime(n) and isprime(n + convert(convert(n,base,10),`+`)), [$1..10^4]); # Robert Israel, Aug 10 2014

Select[Prime[Range[500]],PrimeQ[#+Total[IntegerDigits[#]]]&] (* Harvey P. Dale, Oct 03 2011 *)

select( is(p)=isprime(p+sumdigits(p))&&isprime(p), primes([0,2000])) \\ M. F. Hasler, Aug 08 2014, edited Nov 09 2018

Primes in A047791, i.e., intersection of A047791 and A000040. - M. F. Hasler, Nov 08 2018

A047791 Numbers n such that n plus digit sum of n (A007953) equals a prime.

Original entry on oeis.org

1, 10, 11, 13, 14, 16, 19, 32, 34, 35, 37, 52, 53, 56, 58, 59, 71, 73, 76, 91, 92, 94, 95, 97, 100, 101, 103, 104, 106, 122, 124, 127, 128, 142, 143, 146, 149, 160, 163, 166, 167, 181, 182, 184, 185, 215, 217, 218, 232, 233, 238, 250, 253, 256, 257, 271, 272

Offset: 1

G. L. Honaker, Jr.

			Digit sum of 13 = 1 + 3 = 4 -> 13 + 4 = 17 is prime.

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

Cf. A006378, A107740.

Cf. A007953, A062028, A010051, A107743.

a047791 n = a047791_list !! (n-1)
a047791_list = filter ((== 1) . a010051' . a062028) [1..]
-- Reinhard Zumkeller, Sep 27 2014

Select[Range[272],PrimeQ[#+Total[IntegerDigits[#]]]&] (* Jayanta Basu, May 03 2013 *)

select( is(n)=isprime(n+sumdigits(n)), [1..300]) \\ M. F. Hasler, Nov 08 2018

Complement of A107743.

A062028^(-1)(A000040). - M. F. Hasler, Nov 08 2018

More terms from Larry Reeves (larryr(AT)acm.org), Nov 16 2000

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

A230093 Number of values of k such that k + (sum of digits of k) is n.

Original entry on oeis.org

1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 2, 1

Offset: 0

N. J. A. Sloane, Oct 10 2013

a(n) is the number of times n occurs in A062028.

For n>=1, a(10^n) = a(9*n-1). - Max Alekseyev, Feb 23 2021

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

Cf. A006064, A007953 (sum of digits), A062028 (n + sum of its digits), A004207, A228085, A003052, A176995, A225793, A230094, A055642.

Cf. A107740 (this applied to primes).

a230093 n = length $ filter ((== n) . a062028) [n - 9 * a055642 n .. n]  -- Reinhard Zumkeller, Oct 11 2013

# Maple code for A062028, A230093, A003052, A225793, A230094.
with(LinearAlgebra):
read transforms; # to get digsum
M := 1000; A062028 := Array(0..M); A230093 := Array(0..M);
for n from 0 to M do
   m := n+digsum(n);
   A062028[n] := m;
   if m <= M then A230093[m] := A230093[m]+1; fi;
od:
t1:=[seq(A062028[i],i=0..M)]; # A062028 as list (but incorrect offset 1)
t2:=[seq(A230093[i],i=0..M)]; # A230093 as list, but then a(0) has index 1
# A003052 := COMPl(t1); # COMPl has issues, may be incorrect for M <> 1000
ctmax:=4;
for h from 0 to ctmax do ct[h] := []; od:
for i from 1 to M do
   h := lis2[i];
   if h <= ctmax then ct[h] := [op(ct[h]),i]; fi;
od:
A225793 := ct[1]; A230094 := ct[2]; # A003052 := ct[0]; # see there for better code

Module[{nn=110,a,b,c,d},a=Tally[Table[x+Total[IntegerDigits[x]],{x,0,nn}]];b=a[[All,1]];c={#,0}&/@Complement[Range[nn],b];d=Sort[Join[a,c]];d[[All, 2]]] (* Harvey P. Dale, Jun 12 2019 *)

apply( A230093(n)=sum(i=n>0,min(9*logint(n+!n,10)+8,n\2),sumdigits(n-i)==i), [1..150]) \\ M. F. Hasler, Nov 08 2018

Edited by M. F. Hasler, Nov 08 2018

A007618 a(n) = a(n-1) + sum of digits of a(n-1), a(1) = 5.

Original entry on oeis.org

5, 10, 11, 13, 17, 25, 32, 37, 47, 58, 71, 79, 95, 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

Offset: 1

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

a(n) = A004207(n+5) for n > 52. - Reinhard Zumkeller, Oct 14 2013

a(2) = 10 and a(590) = 10000 are the first two powers of 10 in this sequence; there are no others below a(19017393928) = 1000000000093. Conjecture: the sequence contains infinitely many powers of 10. - Charles R Greathouse IV, Mar 29 2022

N. Agronomof, Problem 4421, L'Intermédiaire des mathématiciens, v. 21 (1914), p. 147. (Mentions sequence starting at 11.) - N. J. A. Sloane, Nov 22 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.
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).

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, A006507, A016096.

a007618 n = a007618_list !! (n-1)
a007618_list = iterate a062028 5  -- Reinhard Zumkeller, Oct 14 2013

from itertools import accumulate
def f(an, _): return an + sum(int(d) for d in str(an))
print(list(accumulate([5]*55, f))) # Michael S. Branicky, May 10 2021

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

cp's OEIS Frontend

A376693 a(n) is the first k such that if x(1) = k and x(i+1) = A062028(x(i)), x(1) to x(n) are all semiprimes but x(n+1) is not.

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A007953 Digital sum (i.e., sum of digits) of n; also called digsum(n).

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A003052 Self numbers or Colombian numbers (numbers that are not of the form m + sum of digits of m for any m).

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

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A092391 a(n) = n + wt(n), where wt(n) = A000120(n) = binary weight of n.

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