A003132 -id:A003132 - cp's OEIS frontend

A309883 Numbers k such that A003132(k^2) = A003132(k), where A003132(n) is the sum of the squares of the digits of n.

0, 1, 10, 35, 100, 152, 350, 377, 452, 539, 709, 1000, 1299, 1398, 1439, 1519, 1520, 1569, 1591, 1679, 1965, 2599, 2838, 3332, 3500, 3598, 3770, 4520, 4586, 4754, 4854, 5390, 5501, 5835, 5857, 6388, 6595, 6735, 6861, 6951, 7090, 7349, 7887, 8395, 9795, 10000, 10056, 10159, 10389, 11055, 11091, 12990, 12999

Offset: 1

Antonio Roldán, Aug 21 2019

If k is in the sequence, then so are k*10^r, r >= 1.

			377^2 = 142129, A003132(377) = 3^2 + 7^2 + 7^2 = 107, A003132(142129) = 1^2 + 4^2 + 2^2 + 1^2 + 2^2 + 9^2 = 107.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A003132, A058369, A070276, A174460, A176670, A178213, A307735, A309884.

[0] cat [k:k in [1..13000]| &+[c^2: c in Intseq(k)] eq &+[c^2: c in Intseq(k^2)]]; // Marius A. Burtea, Aug 24 2019

filter:= proc(n) local t;
  add(t^2, t = convert(n,base,10)) = add(t^2, t = convert(n^2,base,10))
end proc:
select(filter, [$0..20000]); # Robert Israel, Apr 30 2023

digSum[n_] := Total[IntegerDigits[n]^2]; Select[Range[0, 13000], digSum[#] == digSum[#^2] &] (* Amiram Eldar, Aug 22 2019 *)

for(i = 0, 30000, if(norml2(digits(i^2)) == norml2(digits(i)), print1(i, ", ")))

def A003132(n):
    s = 0
    while n > 0:
        s, n = s+(n%10)**2, n//10
    return s
n, a = 0, 0
while n < 50:
    if A003132(a) == A003132(a*a):
        n = n+1
        print(n,a)
    a = a+1 # A.H.M. Smeets, Aug 23 2019

A309884 Numbers k such that A003132(k^3) = A003132(k), where A003132(n) is the sum of the squares of the digits of n.

Original entry on oeis.org

0, 1, 10, 74, 100, 740, 1000, 3488, 7400, 10000, 23658, 30868, 34880, 47508, 48517, 52187, 58947, 59468, 67685, 68058, 74000, 76814, 78368, 78845, 84878, 100000, 108478, 145877, 149217, 163871, 179685, 186884, 188647, 218977, 219878, 236580, 238758, 248967, 278638, 292597, 308680

Offset: 1

Antonio Roldán, Aug 21 2019

If k is in the sequence, then so are k*10^r, with r >= 1.

			74^3 = 405224, A003132(74) = 7^2 + 4^2 = 65, A003132(405224) = 4^2 + 0^2 + 5^2 + 2^2 + 2^2 + 4^2 = 65.

Cf. A003132, A058369, A070276, A174460, A176670, A178213, A307735, A309883.

[0] cat [k:k in [1..310000]| &+[c^2: c in Intseq(k)] eq &+[c^2: c in Intseq(k^3)]]; // Marius A. Burtea, Aug 26 2019

digSum[n_] := Total[IntegerDigits[n]^2]; Select[Range[0, 310000], digSum[#] == digSum[#^3] &] (* Amiram Eldar, Aug 22 2019 *)

for(i = 0, 400000, if(norml2(digits(i^3)) == norml2(digits(i)), print1(i, ", ")))

A342958 a(n) is the least prime that starts a string of exactly n primes p_1, p_2, ... p_n where p_{i+1} = p_i-A003132(p_i), but p_n-A003132(p_n) is not prime.

Original entry on oeis.org

2, 13, 547, 10559, 246349, 20020109, 20020163

Offset: 1

J. M. Bergot and Robert Israel, Mar 31 2021

No further terms < 10^9.

			a(3) = 547 because 547, 547-A003132(547) = 457, and 457-A003132(457) = 367 are prime, but 367-A003132(367) = 273 is not prime.

Cf. A003132, A179549, A342953.

f:= proc(n) option remember; local t, x;
  x:= n - add(t^2, t=convert(n, base, 10));
  if not isprime(x) then 1 else 1+procname(x) fi
end proc:
V:= Vector(7): count:= 0: p:= 1:
while count < 7 do
  p:= nextprime(p); v:= f(p);
  if v <= N and V[v] = 0 then V[v]:= p; count:= count+1 fi
od:
convert(V, list);

A342960 Primes p such that p+A003132(p),(p+A003132(p))+A003132(p+A003132(p)), p-A003132(p), and (p-A003132(p))-A003132(p-A003132(p)) are prime.

Original entry on oeis.org

38377, 70957, 106867, 278177, 278393, 380377, 432199, 435763, 526397, 1093159, 2025577, 2761147, 3068119, 3656129, 3672659, 5649079, 6863173, 7366453, 8083937, 9015863, 9346507, 9497353, 14198467, 15099901, 15467423, 15479273, 16020607, 16437427, 17602547, 18804173, 20020019, 20794141, 22866121

Offset: 1

J. M. Bergot and Robert Israel, Mar 31 2021

The number of digits of p that are not divisible by 3 is divisible by 3.

			a(3) = 106867 is a term because 106867, 106867+A003132(106867) = 107053, 107053+A003132(107053) = 107137, 106867-A003132(106867) = 106681, and 106681-A003132(106681) = 106543 are all prime.

Robert Israel, Table of n, a(n) for n = 1..250

Contained in A179549 and A179550.

Cf. A003132.

filter:= proc(n) local t,x,d;
  if not isprime(n) then return false fi;
  d:= add(t^2, t=convert(n,base,10));
  x:= n+d;
  if not isprime(x) then return false fi;
  if not isprime(x+add(t^2,t=convert(x,base,10))) then return false fi;
  x:= n-d;
  isprime(x) and isprime(x-add(t^2,t=convert(x,base,10)))
end proc:
select(filter, [seq(i,i=3..3*10^7,2)]);

A364682 Number of iterations of the "x -> sum of squares of digits of x" map (A003132) for n to converge to either 0, 1 or the 8-cycle (37,58,89,145,42,20,4,16).

Original entry on oeis.org

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

Offset: 0

Chai Wah Wu, Aug 02 2023

nonn

The initial number counts as iteration 1.

Chai Wah Wu, Table of n, a(n) for n = 0..10000

Cf. A003132, A007770, A039943, A099645, A193995.

def A364682(n):
    c = 1
    while n not in {0,1,37,58,89,145,42,20,4,16}:
        n = sum((0, 1, 4, 9, 16, 25, 36, 49, 64, 81)[ord(d)-48] for d in str(n))
        c += 1
    return c

a(n) <= A193995(n) with equality if and only if n is a Happy number (A007770).
a(n) = 1 if and only if n is in A039943.
a(n) = A099645(n)+1.

A379525 a(1) = 1. For n > 1, if a(n-1) is a novel term, a(n) = A003132(a(n-1)). If a(n-1) has been seen k (>1) times already, a(n) = k*a(n-1).

Original entry on oeis.org

1, 1, 2, 4, 16, 37, 58, 89, 145, 42, 20, 4, 8, 64, 52, 29, 85, 89, 178, 114, 18, 65, 61, 37, 74, 65, 130, 10, 1, 3, 9, 81, 65, 195, 107, 50, 25, 29, 58, 116, 38, 73, 58, 174, 66, 72, 53, 34, 25, 50, 100, 1, 4, 12, 5, 25, 75, 74, 148, 81, 162, 41, 17, 50, 150

Offset: 1

David James Sycamore, Dec 24 2024

2 and 3 appear only once since the only way they can be expressed as the sum of squares is as 2 = 1^2 + 1^2 and 3 = 1^2 + 1^2 + 1^2 (see Example). Maximum number of times prime(k) can appear is A379551(k).

Conjecture: Every positive integer appears in the sequence at least once.

			a(2) = 1^2 = 1 so a(3) = 2 since 1 has occurred twice. This is the only occasion of a(n) = 2 in the sequence. Because a(3) = 2 is a novel term, a(4) = 2^2 = 4, then a(5) = 4^2 = 16.
a(8) = a(18) = 89, so a(19) = 2*89 = 178.
a(28) = 10, a novel term, so a(29) = 1, the 3rd occurrence of 1, so a(30) = 3 (the only occasion of 3 in the sequence).

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^20.

Cf. A003132, A379551.

nn = 120; c[_] := 0; a[1] = j = 1;
Do[If[c[j] == 0,
    k = Total[IntegerDigits[j]^2]; c[j]++,
    k = ++c[j]*j ];
  Set[{a[n], j}, {k, k}], {n, 2, nn}];
Array[a, nn] (* Michael De Vlieger, Dec 25 2024 *)

A099652 Largest number arising if f[x]=1+A003132(x) function is iterated until steady state reached. Compare with A099648.

Original entry on oeis.org

107, 107, 107, 107, 107, 107, 107, 107, 118, 107, 107, 107, 107, 107, 107, 107, 107, 107, 107, 107, 107, 118, 107, 107, 107, 107, 107, 118, 107, 107, 107, 107, 107, 107, 35, 53, 107, 107, 107, 107, 107, 107, 107, 107, 107, 53, 107, 107, 146, 107, 107, 107, 53

Offset: 1

Labos Elemer, Nov 12 2004

			n=32: list={32,14,18,66,73,59,107,51,27,54,42,21,6,37,59,107,51,...},
max[list]=a[32]=107>initial-value;
Also:lengths of transient=5,of cycle=0 (see in A099646).

Cf. A003132, A099648, A099646, A099649.

ed[x_] :=IntegerDigits[x];func[x_] :=Apply[Plus, ed[x]^2]+1; Table[Max[NestList[func, w, 200]], {w, 1, 150}]

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

Original entry on oeis.org

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

A055012 Sum of cubes of the digits of n written in base 10.

Original entry on oeis.org

0, 1, 8, 27, 64, 125, 216, 343, 512, 729, 1, 2, 9, 28, 65, 126, 217, 344, 513, 730, 8, 9, 16, 35, 72, 133, 224, 351, 520, 737, 27, 28, 35, 54, 91, 152, 243, 370, 539, 756, 64, 65, 72, 91, 128, 189, 280, 407, 576, 793, 125, 126, 133, 152, 189, 250, 341, 468, 637, 854

Offset: 0

Henry Bottomley, May 31 2000

For n > 1999, a(n) < n. The iteration of this map on n either stops at a fixed point (A046197) or has a period of 2 or 3: {55,250,133}, {136,244}, {160,217,352}, or {919,1459}. - T. D. Noe, Jul 17 2007

A165330 and A165331 give the final value and the number of steps when iterating until a fixed point or cycle is reached. - Reinhard Zumkeller, Sep 17 2009

T. D. Noe, Table of n, a(n) for n = 0..11000
K. Iséki, A problem of number theory, Proc. Japan Academy 36 (1960), 578-583.
B. M. Stewart, Sums of functions of digits, Canad. J. Math., 12 (1960), 374-389.
Index entries for Colombian or self numbers and related sequences

Cf. A003132, A031179.
Cf. A046197 Fixed points; A046459: integers equal to the sum of the digits of their cubes; A072884: 3rd-order digital invariants: the sum of the cubes of the digits of n equals some number k and the sum of the cubes of the digits of k equals n; A164883: cubes with the property that the sum of the cubes of the digits is also a cube.
Cf. A003132, A007953, A055017, A076313, A076314, A165340.

[0] cat [&+[d^3: d in Intseq(n)]: n in [1..60]]; // Bruno Berselli, Feb 01 2013

A055012 := proc(n)
        add(d^3,d=convert(n,base,10)) ;
end proc: # R. J. Mathar, Dec 15 2011

Total/@((IntegerDigits/@Range[0,60])^3) (* Harvey P. Dale, Jan 27 2012 *)
Table[Sum[DigitCount[n][[i]] i^3, {i, 9}], {n, 0, 60}] (* Bruno Berselli, Feb 01 2013 *)

A055012(n)=sum(i=1,#n=digits(n),n[i]^3) \\ Charles R Greathouse IV, Jul 01 2013

def a(n): return sum(map(lambda x: x*x*x, map(int, str(n))))
print([a(n) for n in range(60)]) # Michael S. Branicky, Jul 13 2022

a(n) = Sum_{k>=1} (floor(n/10^k) - 10*floor(n/10^(k+1)))^3. - Hieronymus Fischer, Jun 25 2007
a(10n+k) = a(n) + k^3, 0 <= k < 10. - Hieronymus Fischer, Jun 25 2007
From Reinhard Zumkeller, Sep 17 2009: (Start)
a(n) <= 729*A055642(n);
a(A165370(n)) = n and a(m) <> n for m < A165370(n);
a(A031179(n)) = A031179(n);
a(a(A165336(n))) = A165336(n) or a(a(a(A165336(n)))) = A165336(n). (End)
G.f. g(x) = Sum_{k>=0} (1-x^(10^k))*(x^(10^k)+8*x^(2*10^k)+27*x^(3*10^k)+64*x^(4*10^k)+125*x^(5*10^k)+216*x^(6*10^k)+343*x^(7*10^k)+512*x^(8*10^k)+729*x^(9*10^k))/((1-x)*(1-x^(10^(k+1))))
satisfies
g(x) = (x+8*x^2+27*x^3+64*x^4+125*x^5+216*x^6+343*x^7+512*x^8+729*x^9)/(1-x^10) + (1-x^10)*g(x^10)/(1-x). - Robert Israel, Jan 26 2017

Edited by M. F. Hasler, Apr 12 2015

Iséki and Stewart links added by Don Knuth, Sep 07 2015

A051885 Smallest number whose sum of digits is n.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 19, 29, 39, 49, 59, 69, 79, 89, 99, 199, 299, 399, 499, 599, 699, 799, 899, 999, 1999, 2999, 3999, 4999, 5999, 6999, 7999, 8999, 9999, 19999, 29999, 39999, 49999, 59999, 69999, 79999, 89999, 99999, 199999, 299999, 399999, 499999

Offset: 0

Felice Russo, Dec 15 1999

This is also the list of lunar triangular numbers: A087052 with duplicates removed. - N. J. A. Sloane, Jan 25 2011
Numbers n such that A061486(n) = n. - Amarnath Murthy, May 06 2001
The product of digits incremented by 1 is the same as the number incremented by 1. If a(n) = abcd...(a,b,c,d, etc. are digits of a(n)) {a(n) + 1} = (a+1)*(b+1)(c+1)*(d+1)*..., e.g., 299 + 1 = (2+1)*(9+1)*(9+1) = 300. - Amarnath Murthy, Jul 29 2003
A138471(a(n)) = 0. - Reinhard Zumkeller, Mar 19 2008
a(n+1) = A108971(A179988(n)). - Reinhard Zumkeller, Aug 09 2010, Jul 10 2011
Positions of records in A003132: A080151(n) = A003132(a(n)). - Reinhard Zumkeller, Jul 10 2011
a(n) = A242614(n,1). - Reinhard Zumkeller, Jul 16 2014
A254524(a(n)) = 1. - Reinhard Zumkeller, Oct 09 2015
The slowest strictly increasing sequence of nonnegative integers such that, for any two terms, calculating the difference of their decimal representations requires no borrowing. - Rick L. Shepherd, Aug 11 2017

Iain Fox, Table of n, a(n) for n = 0..9000 (first 101 terms from Reinhard Zumkeller)
D. Applegate, M. LeBrun and N. J. A. Sloane, Dismal Arithmetic, arXiv:1107.1130 [math.NT], 2011. [Note: we have now changed the name from "dismal arithmetic" to "lunar arithmetic" - the old name was too depressing]
A. Murthy, Exploring some new ideas on Smarandache type sets, functions and sequences, Smarandache Notions Journal Vol. 11 N. 1-2-3 Spring 2000.
Index entries for sequences related to dismal (or lunar) arithmetic
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,10,-10).

Cf. A061104, A061105, A061486, A007953, A067043, A087052.
Numbers of form i*b^j-1 (i=1..b-1, j >= 0) for bases b = 2 through 9: A000225, A062318, A180516, A181287, A181288, A181303, A165804, A140576. - N. J. A. Sloane, Jan 25 2011
Cf. A002283.
Cf. A254524.

a051885 n = (m + 1) * 10^n' - 1 where (n',m) = divMod n 9
-- Reinhard Zumkeller, Jul 10 2011

Magma
```
[i*10^j-1: i in [1..9], j in [0..5]];
```

b:=10; t1:=[]; for j from 0 to 15 do for i from 1 to b-1 do t1:=[op(t1), i*b^j-1]; od: od: t1; # N. J. A. Sloane, Jan 25 2011

a[n_] := (Mod[n, 9] + 1)*10^Floor[n/9] - 1; Table[a[n], {n, 0, 49}](* Jean-François Alcover, Dec 01 2011, after Henry Bottomley *)

A051885(n) = (n%9+1)*10^(n\9)-1  \\ M. F. Hasler, Jun 17 2012

first(n) = Vec(x*(x^2 + x + 1)*(x^6 + x^3 + 1)/((x - 1)*(10*x^9 - 1)) + O(x^n), -n) \\ Iain Fox, Dec 30 2017

def A051885(n): return ((n % 9)+1)*10**(n//9)-1 # Chai Wah Wu, Apr 04 2021

These are the numbers i*10^j-1 (i=1..9, j >= 0). - N. J. A. Sloane, Jan 25 2011
a(n) = ((n mod 9) + 1)*10^floor(n/9) - 1 = a(n-1) + 10^floor((n-1)/9). - Henry Bottomley, Apr 24 2001
a(n) = A037124(n+1) - 1. - Reinhard Zumkeller, Jan 03 2008, Jul 10 2011
G.f.: x*(x^2+x+1)*(x^6+x^3+1) / ((x-1)*(10*x^9-1)). - Colin Barker, Feb 01 2013

More terms from James Sellers, Dec 16 1999

Offset fixed by Reinhard Zumkeller, Jul 10 2011

cp's OEIS Frontend

A309883 Numbers k such that A003132(k^2) = A003132(k), where A003132(n) is the sum of the squares of the digits of n.

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A309884 Numbers k such that A003132(k^3) = A003132(k), where A003132(n) is the sum of the squares of the digits of n.

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A342958 a(n) is the least prime that starts a string of exactly n primes p_1, p_2, ... p_n where p_{i+1} = p_i-A003132(p_i), but p_n-A003132(p_n) is not prime.

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A342960 Primes p such that p+A003132(p),(p+A003132(p))+A003132(p+A003132(p)), p-A003132(p), and (p-A003132(p))-A003132(p-A003132(p)) are prime.

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A364682 Number of iterations of the "x -> sum of squares of digits of x" map (A003132) for n to converge to either 0, 1 or the 8-cycle (37,58,89,145,42,20,4,16).

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A379525 a(1) = 1. For n > 1, if a(n-1) is a novel term, a(n) = A003132(a(n-1)). If a(n-1) has been seen k (>1) times already, a(n) = k*a(n-1).

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A099652 Largest number arising if f[x]=1+A003132(x) function is iterated until steady state reached. Compare with A099648.

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

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