A058369 -id:A058369 - 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

A004159 Sum of digits of n^2.

Original entry on oeis.org

0, 1, 4, 9, 7, 7, 9, 13, 10, 9, 1, 4, 9, 16, 16, 9, 13, 19, 9, 10, 4, 9, 16, 16, 18, 13, 19, 18, 19, 13, 9, 16, 7, 18, 13, 10, 18, 19, 13, 9, 7, 16, 18, 22, 19, 9, 10, 13, 9, 7, 7, 9, 13, 19, 18, 10, 13, 18, 16, 16, 9, 13, 19, 27, 19, 13, 18, 25, 16, 18, 13, 10, 18, 19, 22, 18, 25, 25, 18, 13

Offset: 0

N. J. A. Sloane

If 3|n then 9|a(n); otherwise, a(n) == 1 (mod 3). - Jon E. Schoenfield, Jun 30 2018

			Trajectories under the map x -> a(x):
1 ->  1 ->  1 ->  1 ->  1 ->  1 ->  1 ->  1 ->  1 -> ...
2 ->  4 ->  7 -> 13 -> 16 -> 13 -> 16 -> 13 -> 16 -> ...
3 ->  9 ->  9 ->  9 ->  9 ->  9 ->  9 ->  9 ->  9 -> ...
4 ->  7 -> 13 -> 16 -> 13 -> 16 -> 13 -> 16 -> 13 -> ...
5 ->  7 -> 13 -> 16 -> 13 -> 16 -> 13 -> 16 -> 13 -> ...
6 ->  9 ->  9 ->  9 ->  9 ->  9 ->  9 ->  9 ->  9 -> ...
7 -> 13 -> 16 -> 13 -> 16 -> 13 -> 16 -> 13 -> 16 -> ...
- _R. J. Mathar_, Jul 08 2012

Zak Seidov, Table of n, a(n) for n = 0..10000
A. S. Besicovitch, The asymptotic distribution of the numerals in the decimal representation of the squares of the natural numbers, Mathematische Zeitschrift 39 (1934), pp. 146-156.
H. Davenport and P. Erdős, Note on normal decimals, Canadian Journal of Mathematics 4 (1952), pp. 58-63.
Michael Drmota, Christian Mauduit and Joël Rivat, The sum-of-digits function of polynomial sequences, J. Lond. Math. Soc. (2) 84(2011), no. 1, 81--102. MR2819691 (2012f:11193)
Bernt Lindström, On the binary digits of a power, Journal of Number Theory, Volume 65, Issue 2, August 1997, Pages 321-324.
Christian Mauduit and Joël Rivat, La somme des chiffres des carrés, Acta Mathem. 203 (1) (2009) 107-148. MR2545827 (2010j:11119).
H. I. Okagbue, M. O. Adamu, S. A. Iyase and A. A. Opanuga, Sequence of Integers Generated by Summing the Digits of their Squares, Indian Journal of Science and Technology, Vol 8(15), DOI: 10.17485/ijst/2015/v8i15/69912, July 2015.
K. B. Stolarsky, The binary digits of a power, Proc. Amer. Math. Soc. 71 (1978), 1-5.

Cf. A007953, A159918, A056691, A268226.
Cf. A240752 (first differences), A071317 (partial sums).
Cf. A062685 (smallest square with digit sum n, or 0 if no such square exists).

a004159 = a007953 . a000290  -- Reinhard Zumkeller, Apr 12 2014

read("transforms"):
A004159 := proc(n)
        digsum(n^2) ;
end proc: # R. J. Mathar, Jul 08 2012

a004159[n_Integer] := Apply[Plus, IntegerDigits[n^2]]; Table[
a004159[n], {n, 0, 100}] (* Michael De Vlieger, Jul 21 2014 *)
Total[IntegerDigits[#]]&/@(Range[0,100]^2) (* Harvey P. Dale, Feb 03 2019 *)

A004159(n)=sumdigits(n^2) \\ M. F. Hasler, Sep 23 2014

def A004159(n):
    return sum(int(d) for d in str(n*n)) # Chai Wah Wu, Sep 03 2014

a(n) = A007953(A000290(n)); a(A058369(n)) = A007953(A058369(n)). - Reinhard Zumkeller, Apr 25 2009

a(10n) = a(n). If n > 1 is not a multiple of 10, then a(n)=4 iff n = 10^k+1 = A062397(k), a(n)=7 iff n is in A215614={4, 5, 32, 49, 149, 1049}, and else a(n) >= 9. - M. F. Hasler, Sep 23 2014

A077436 Let B(n) be the sum of binary digits of n. This sequence contains n such that B(n) = B(n^2).

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 7, 8, 12, 14, 15, 16, 24, 28, 30, 31, 32, 48, 56, 60, 62, 63, 64, 79, 91, 96, 112, 120, 124, 126, 127, 128, 157, 158, 159, 182, 183, 187, 192, 224, 240, 248, 252, 254, 255, 256, 279, 287, 314, 316, 317, 318, 319, 351, 364, 365, 366, 374, 375, 379, 384

Offset: 1

Giuseppe Melfi, Nov 30 2002

Superset of A023758.

Hare, Laishram, & Stoll show that this sequence contains infinitely many odd numbers. In particular for each k in {12, 13, 16, 17, 18, 19, 20, ...} there are infinitely many terms in this sequence with binary digit sum k. - Charles R Greathouse IV, Aug 25 2015

			The element 79 belongs to the sequence because 79=(1001111) and 79^2=(1100001100001), so B(79)=B(79^2)

Reinhard Zumkeller, Table of n, a(n) for n = 1..10476, all terms <= 2^20
Karam Aloui, Damien Jamet, Hajime Kaneko, Steffen Kopecki, Pierre Popoli, and Thomas Stoll, On the binary digits of n and n^2, arXiv:2203.05451 [math.NT], 2022.
K. G. Hare, S. Laishram, and T. Stoll, The sum of digits of n and n^2, International Journal of Number Theory 7:7 (2011), pp. 1737-1752.
Giuseppe Melfi, On simultaneous binary expansions of n and n^2, arXiv:math/0402458 [math.NT], 2004.
Giuseppe Melfi, Su alcune successioni di interi (English with an Italian title)

Cf. A058369, A000120, A000290, A083567.
Cf. A211676 (number of n-bit numbers in this sequence).
A261586 is a subsequence. Subsequence of A352084.

import Data.List (elemIndices)
import Data.Function (on)
a077436 n = a077436_list !! (n-1)
a077436_list = elemIndices 0
   $ zipWith ((-) `on` a000120) [0..] a000290_list
-- Reinhard Zumkeller, Apr 12 2011

[n: n in [0..400] | &+Intseq(n, 2) eq &+Intseq(n^2, 2)]; // Vincenzo Librandi, Aug 30 2015

select(t -> convert(convert(t,base,2),`+`) = convert(convert(t^2,base,2),`+`), [$0..1000]); # Robert Israel, Aug 27 2015

t={}; Do[If[DigitCount[n, 2, 1] == DigitCount[n^2, 2, 1], AppendTo[t, n]], {n, 0, 364}]; t
f[n_] := Total@ IntegerDigits[n, 2]; Select[Range[0, 384], f@ # == f[#^2] &] (* Michael De Vlieger, Aug 27 2015 *)

is(n)=hammingweight(n)==hammingweight(n^2) \\ Charles R Greathouse IV, Aug 25 2015

def ok(n): return bin(n).count('1') == bin(n**2).count('1')
print([m for m in range(400) if ok(m)]) # Michael S. Branicky, Mar 11 2022

A159918(a(n)) = A000120(a(n)). - Reinhard Zumkeller, Apr 25 2009

Initial 0 added by Reinhard Zumkeller, Apr 28 2012, Apr 12 2011

A254066 Primitive numbers n such that the sums of the digits of n and n^2 coincide.

Original entry on oeis.org

1, 9, 18, 19, 45, 46, 55, 99, 145, 189, 198, 199, 289, 351, 361, 369, 379, 388, 451, 459, 468, 495, 496, 558, 559, 568, 585, 595, 639, 729, 739, 775, 838, 855, 954, 955, 999, 1098, 1099, 1179, 1188, 1189, 1198, 1269, 1468, 1485, 1494, 1495, 1585, 1738, 1747

Offset: 1

Nikhil Mahajan, Jan 25 2015

Members of A058369 not congruent to 0 (mod 10).
This sequence is to A058369 what A114135 is to A111434.
Hare, Laishram, & Stoll show that this sequence is infinite. In particular for each k in {846, 847, 855, 856, 864, 865, 873, ...} there are infinitely many terms in this sequence with digit sum k. - Charles R Greathouse IV, Aug 25 2015

			9 is in the sequence because the digit sum of 9^2 = 81 is 9.
18 is in the sequence because the digit sum of 18^2 = 324 is 9, same as the digit sum of 18.

Nikhil Mahajan, Table of n, a(n) for n = 1..10000
K. G. Hare, S. Laishram, and T. Stoll, The sum of digits of n and n^2, International Journal of Number Theory 7:7 (2011), pp. 1737-1752.

Cf. A058369, A114135, A111434.

Subsequence of A090570.

[n: n in [1..1000] | &+Intseq(n) eq &+Intseq(n^2) and not IsZero(n mod 10)]; // Bruno Berselli, Jan 29 2015

Select[Range[1000],!Divisible[#,10]&&Total[IntegerDigits[#]] == Total[ IntegerDigits[#^2]]&] (* Harvey P. Dale, Dec 27 2015 *)

is(n)=sumdigits(n)==sumdigits(n^2) \\ Charles R Greathouse IV, Aug 25 2015

list(lim)=my(v=List()); forstep(n=1,lim,[8, 9, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 9], if(sumdigits(n)==sumdigits(n^2), listput(v,n))); Vec(v) \\ Charles R Greathouse IV, Aug 26 2015

[n for n in [0..1000] if sum(n.digits())==sum((n^2).digits()) and n%10!=0] # Tom Edgar, Jan 27 2015

More terms from Harvey P. Dale, Dec 27 2015

A111434 Numbers k such that the sums of the digits of k, k^2 and k^3 coincide.

Original entry on oeis.org

0, 1, 10, 100, 468, 585, 1000, 4680, 5850, 5851, 5868, 10000, 28845, 46800, 58500, 58510, 58680, 58968, 100000, 288450, 468000, 585000, 585100, 586800, 589680, 1000000, 2884500, 4680000, 5850000, 5851000, 5868000, 5896800, 10000000

Offset: 1

Giovanni Resta, Nov 21 2005

The sequence is clearly infinite, since we can add trailing zeros. Is the subset of values not ending in 0 infinite too (see A114135)?

			468 is in the sequence since 468^2 = 219024 and 468^3 = 102503232 and we have 18 = 4+6+8 = 2+1+9+0+2+4 = 1+0+2+5+0+3+2+3+2.
5851 is in the sequence because 5851, 34234201 (= 5851^2) and 200304310051 (=5851^3) all have digital sum 19.

David A. Corneth, Table of n, a(n) for n = 1..1124 (using the b-file in A114135).

Cf. A011557, A058369, A070276.

s:=proc(n) local nn: nn:=convert(n,base,10): sum(nn[j],j=1..nops(nn)): end: a:=proc(n) if s(n)=s(n^2) and s(n)=s(n^3) then n else fi end: seq(a(n),n=0..1000000); # Emeric Deutsch, May 13 2006

SumOfDig[n_]:=Apply[Plus, IntegerDigits[n]]; Do[s=SumOfDig[n]; If[s==SumOfDig[n^2] && s==SumOfDig[n^3], Print[n]], {n, 10^6}]
Select[Range[0,10000000],Length[Union[Total/@IntegerDigits[{#,#^2,#^3}]]] == 1&] (* Harvey P. Dale, Apr 26 2014 *)

b-file Corrected by David A. Corneth, Jul 22 2021

A058852 Palindromes n such that n and n^2 have same digit sum.

Original entry on oeis.org

0, 1, 9, 55, 99, 585, 595, 838, 999, 5995, 8668, 9999, 45954, 48384, 55755, 56665, 59895, 59995, 64846, 65656, 77977, 86968, 88488, 89398, 97479, 97579, 99099, 99199, 99999, 158851, 176671, 509905, 549945, 558855, 594495, 599995, 779977

Offset: 1

Patrick De Geest, Dec 15 2000

R. J. Mathar, Table of n, a(n) for n = 1..399

Cf. A058369, A058370, A007953.

read("transforms") :
n := 1;
for i from 1 do
    p := A002113(i) ;
    if digsum(p) = digsum(p^2) then
        printf("%d %d\n",n,p) ;
        n := n+1 ;
    end if;
end do: # R. J. Mathar, Sep 09 2015

Select[Range[0,78*10^4],PalindromeQ[#]&&Total[IntegerDigits[#]] == Total[ IntegerDigits[ #^2]]&] (* Harvey P. Dale, Sep 26 2021 *)

A292787 For n > 1, a(n) = least positive k, not a power of n, such that the digital sum of k in base n equals the digital sum of k^2 in base n.

Original entry on oeis.org

3, 2, 3, 4, 5, 3, 7, 8, 9, 5, 11, 4, 13, 7, 6, 16, 17, 9, 19, 5, 7, 11, 23, 9, 25, 13, 27, 8, 29, 6, 31, 32, 12, 17, 15, 9, 37, 19, 13, 16, 41, 7, 43, 12, 10, 23, 47, 16, 49, 25, 18, 13, 53, 27, 11, 8, 19, 29, 59, 16, 61, 31, 28, 64, 26, 12, 67, 17, 24, 15, 71

Offset: 2

Rémy Sigrist, Sep 23 2017

For n > 2, a(n) <= n-1 (and the sum of digits of a(n)^2 in base n equals a(n)).
The term a(10) = 9 belongs to A058369.
For any n > 1 and k >= 0:
- let d_n(k) be the digital sum of k in base n,
- we have d_n(1) = 1 and d_n(n-1) = d_n((n-1)^2),
- for any k such that 0 <= k <= n, we have d_n(k^2) - d_n(k) = d_n((n-k)^2) - d_n(n-k),
- for any even n > 2, d_n(n/2) != d_n((n/2)^2),
- hence, for any n > 2, either a(n) < n/2 or a(n) = n-1,
- and the scatterplot of the sequence (for n > 2) has only points in the region y < x/2 and on the line y = x-1.
Apparently (see colorized scatterplot in Links section):
- for any k > 0, a(2^k + 1) = 2^k,
- if n is odd and not of the form 2^k + 1, then a(n) <= (n-1)/2.
See also A292788 for a similar sequence involving cubes instead of squares.

			For n = 8:
- 1 is a power of 8,
- d_8(2) = 2 and d_8(2^2) = 4,
- d_8(3) = 3 and d_8(3^2) = 2,
- d_8(4) = 4 and d_8(4^2) = 2,
- d_8(5) = 5 and d_8(5^2) = 4,
- d_8(6) = 6 and d_8(6^2) = 8,
- d_8(7) = 7 and d_8(7^2) = 7,
- hence a(8) = 7.

Rémy Sigrist, Scatterplot of the sequence for n=2..10000
Rémy Sigrist, Colorized scatterplot of the sequence for n=2..10000

Cf. A058369, A292788.

With[{kk = 10^3}, Table[SelectFirst[Complement[Range[2, kk], n^Range@ Floor@ Log[n, kk]], Total@ IntegerDigits[#, n] == Total@ IntegerDigits[#^2, n] &] /. k_ /; MissingQ@ k -> -1, {n, 2, 72}]] (* Michael De Vlieger, Sep 24 2017 *)

a(n) = my (p=1); for (k=1, oo, if (k==p, p*=n, if (sumdigits(k,n) == sumdigits(k^2,n), return (k))))

A049343 Numbers m such that 2m and m^2 have same digit sum.

Original entry on oeis.org

0, 2, 9, 11, 18, 20, 29, 38, 45, 47, 90, 99, 101, 110, 119, 144, 146, 180, 182, 189, 198, 200, 245, 290, 299, 335, 344, 351, 362, 369, 380, 398, 450, 452, 459, 461, 468, 470, 479, 488, 495, 497, 639, 729, 794, 839, 848, 900, 929, 954, 990, 999

Offset: 1

R. K. Guy

An easy way to prove that this sequence is infinite is to observe that it contains all numbers of the form 10^k+1. - Stefan Steinerberger, Mar 31 2006

For n>0: digital root (A010888) of 2n or n^2 is either 4 or 9. - Reinhard Zumkeller, Oct 01 2007

Problem 117 in Loren Larson's translation of Paul Vaderlind's book.

Reinhard Zumkeller and Harvey P. Dale, Table of n, a(n) for n = 1..1000 (first 101 terms from Zumkeller)

Cf. A058369, A077436 (binary). - Reinhard Zumkeller, Apr 03 2011

import Data.List (elemIndices)
import Data.Function (on)
a049343 n = a049343_list !! (n-1)
a049343_list = elemIndices 0
   $ zipWith ((-) `on` a007953) a005843_list a000290_list
-- Reinhard Zumkeller, Apr 03 2011

[n: n in [0..1000] | &+Intseq(2*n) eq &+Intseq(n^2)]; // Vincenzo Librandi, Nov 17 2015

Select[Range[0, 1000], Sum[DigitCount[2# ][[i]]*i, {i, 1, 9}] == Sum[DigitCount[ #^2][[i]]*i, {i, 1, 9}] &] (* Stefan Steinerberger, Mar 31 2006 *)
Select[Range[0,1000],Total[IntegerDigits[2#]]==Total[ IntegerDigits[ #^2]]&] (* Harvey P. Dale, Sep 25 2012 *)

A007953(A005843(a(n))) = A007953(A000290(a(n))). - Reinhard Zumkeller, Oct 01 2007

cp's OEIS Frontend

A147523 a(n) = number of n-digit terms in A058369.

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

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A004159 Sum of digits of n^2.

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A077436 Let B(n) be the sum of binary digits of n. This sequence contains n such that B(n) = B(n^2).

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A254066 Primitive numbers n such that the sums of the digits of n and n^2 coincide.

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