A055642 -id:A055642 - cp's OEIS frontend

A004427 Arithmetic mean of digits of n (rounded up).

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

Offset: 0

N. J. A. Sloane

a(100)=1 is the first value that differs from the variant "... rounded to the nearest integer". - M. F. Hasler, May 10 2015

Cf. A178358, A178359, A178361, A178362, ..., A178369, A178401, A178402, A178403, A178404, A178405.

Cf. A000040, A004426, A007953, A055642, A061383, A074462.

Ceiling[Mean[IntegerDigits[#]]]&/@Range[0,110] (* Harvey P. Dale, Aug 29 2014 *)

A004427(n)=ceil(sum(i=1, #n=digits(n), n[i])/#n) \\ ...Vecsmall(Str(n))...-48 is a little faster. \\ M. F. Hasler, May 10 2015

From Reinhard Zumkeller, May 27 2010: (Start)
a(n) = ceiling(A007953(n)/A055642(n)); a(A000040(n)) = A074462(n);
A004426(n) <= a(n) with equality for n in A061383;
a(A178361(n)) = 1; a(A178362(n)) = 2; a(A178363(n)) = 3; a(A178364(n)) = 4; a(A178365(n)) = 5; a(A178366(n)) = 6; a(A178367(n)) = 7; a(A178368(n)) = 8; a(A178369(n)) = 9. (End)

A032810 Numbers using only digits 2 and 3.

Original entry on oeis.org

2, 3, 22, 23, 32, 33, 222, 223, 232, 233, 322, 323, 332, 333, 2222, 2223, 2232, 2233, 2322, 2323, 2332, 2333, 3222, 3223, 3232, 3233, 3322, 3323, 3332, 3333, 22222, 22223, 22232, 22233, 22322, 22323, 22332, 22333, 23222, 23223

Offset: 1

Clark Kimberling

Identical to A007931 with substitution of digits 2 -> 3, 1 -> 2, i.e., application of the function A048379 or A256079(n) = n + A002275(A055642(n)). - M. F. Hasler, Mar 21 2015

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for 10-automatic sequences.

Cf. A020458, A143967, A248907 (permutation).

Cf. A032804-A032816 (in other bases), A007088 (digits 0 & 1), A007931 (digits 1 & 2), A032834 (digits 3 & 4), A256290 (digits 4 & 5), A256291 (digits 5 & 6), A256292 (digits 6 & 7), A256340 (digits 7 & 8), A256341 (digits 8 & 9).

a032810 = f 0 . (+ 1) where
   f y 1 = a004086 y
   f y x = f (10 * y + m + 2) x' where (x', m) = divMod x 2
-- Reinhard Zumkeller, Mar 18 2015

[n: n in [1..24000] | Set(Intseq(n)) subset {2, 3}]; // Vincenzo Librandi, May 27 2012

[n eq 1 select 2 else IsOdd(n) select 10*Self(Floor(n/2))+2 else Self(n-1)+1: n in [1..40]]; // Bruno Berselli, May 27 2012

Flatten[Table[FromDigits[#,10]&/@Tuples[{2,3},n],{n,5}]] (* Vincenzo Librandi, May 27 2012 *)

A032810(n)=vector(#n=binary(n+1)[2..-1],i,10^(#n-i))*n~+10^#n\9*2 \\ M. F. Hasler, Mar 26 2015

def A032810(n): return int(bin(n+1)[3:])+(10**((n+1).bit_length()-1)-1<<1)//9 # Chai Wah Wu, Jul 15 2023

a(n) = f(n+1, 0) with f(n, x) = if n=1 then A004086(x) else f(floor(n/2), 10*x + 2 + n mod 2). - Reinhard Zumkeller, Sep 06 2008
a(n) is Theta(n^(log_2 10)); there are about n^(log_10 2) members of this sequence up to n. - Charles R Greathouse IV, Mar 18 2010
a(n) = A007931(n) + A002275(A000523(n+1)). A055642(a(n)) = A000523(n+1). - M. F. Hasler, Mar 21 2015

A047726 Number of different numbers that are formed by permuting digits of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 3, 3, 6

Offset: 1

N. J. A. Sloane

The minimum value of a(A171102(n)) is 10*9!. - Altug Alkan, Jul 08 2016

			From 102 we get 102, 120, 210, 201, 12 and 21, so a(102)=6.
From 33950 with 5 digits, one '0', two '3', one '5' and one '9', we get 5! / (1! * 2! * 1! * 1!) = 60 different numbers and a(33950) = 60.  - _Bernard Schott_, Oct 20 2019

A. Dunigan AtLee, Table of n, a(n) for n = 1..100000.

Cf. A055098. Identical to A043537 and A043562 for n<100.

Cf. A179239. - Aaron Dunigan AtLee, Jul 14 2010

import Data.List (permutations, nub)
a047726 n = length $ nub $ permutations $ show n
-- Reinhard Zumkeller, Jul 26 2011

f:= proc(n) local L;
  L:= convert(n,base,10);
  nops(L)!/mul(numboccur(i,L)!,i=0..9);
end proc:
map(f, [$1..1000]); # Robert Israel, Jul 08 2016

pd[n_]:=Module[{p=Permutations[IntegerDigits[n]]},Length[Union [FromDigits/@p]]]; pd/@Range[120]  (* Harvey P. Dale, Mar 22 2011 *)

a(n)=n=eval(Vec(Str(n)));(#n)!/prod(i=0,9,sum(j=1,#n,n[j]==i)!) \\ Charles R Greathouse IV, Sep 29 2011

A047726(n)={local(c=Vec(0,10)); apply(d->c[d+1]++, digits(n)); logint(n*10,10)!/prod(i=1,10,c[i]!)} \\ M. F. Hasler, Oct 18 2019

a(n) << n / (log_10 n)^4.5 by Stirling's approximation. - Charles R Greathouse IV, Sep 29 2011

a(n) = A000142(A055642(n))/Product_{k=0..9} A000142(A100910(n,k)). - Robert Israel, Jul 08 2016

Corrected by Henry Bottomley, Apr 19 2000

A160093 Number of digits in n, excluding any trailing zeros.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 3, 3, 3, 3, 3

Offset: 1

Anonymous, May 01 2009

			a(1060000) = 3 because discarding the trailing zeros from 1060000 leaves 106, which is a 3-digit number.

Indranil Ghosh, Table of n, a(n) for n = 1..10000

Cf. A004151, A054899, A055640, A055641, A055642, A102669, A122840, A122841, A160094, A196563, A196564.

lnzd[n_]:=Module[{spl=Last[Split[IntegerDigits[n]]]},If[!MemberQ[ spl,0], IntegerLength[n], IntegerLength[n]-Length[spl]]]; Array[lnzd,110] (* Harvey P. Dale, Jun 05 2013 *)
Table[IntegerLength[n] - IntegerExponent[n, 10], {n, 100}] (* Amiram Eldar, Sep 14 2020 *)

a(n)=if(n==0,1,#digits(n/10^valuation(n,10))) \\ Joerg Arndt, Jan 11 2017

a(n)=logint(n,10)+1-valuation(n,10) \\ Charles R Greathouse IV, Jan 12 2017

def A160093(n):
     return len(str(int(str(n)[::-1]))) # Indranil Ghosh, Jan 11 2017

From Hieronymus Fischer, Jun 08 2012: (Start)
With m = floor(log_10(n)), frac(x) = x-floor(x):
a(n) = 1 + Sum_{j=0..m} ceiling(frac(n/10^j)).
a(n) = 1 - Sum_{j=1..m} (floor(-frac(n/10^j))).
a(n)= A055642(n) + A054899(n-1) - A054899(n).
G.f.: (x/(1-x)) + (1/(1-x))*Sum_{j>0} x^(10^j+1)*(1 - x^(10^j-1))/(1-x^10^j). (End)
a(n) = A055642(A004086(n)). - Indranil Ghosh, Jan 11 2017
a(n) = A055642(A004151(n)). - Amiram Eldar, Sep 14 2020

Simpler definition and changed example from Jon E. Schoenfield, Feb 15 2014

A034887 Number of digits in 2^n.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 6, 6, 6, 7, 7, 7, 7, 8, 8, 8, 9, 9, 9, 10, 10, 10, 10, 11, 11, 11, 12, 12, 12, 13, 13, 13, 13, 14, 14, 14, 15, 15, 15, 16, 16, 16, 16, 17, 17, 17, 18, 18, 18, 19, 19, 19, 19, 20, 20, 20, 21, 21, 21, 22, 22, 22

Offset: 0

N. J. A. Sloane

The sequence consists of the positive integers, each appearing 3 or 4 times. - M. F. Hasler, Oct 08 2016

T. D. Noe, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Mersenne Number

Cf. A055253, A055254.
Cf. A000079, A055642.
See A125117 for the sequence of first differences.

[#Intseq(2^n): n in [0..100] ]; // Vincenzo Librandi, Jun 23 2015

seq(floor(n*ln(2)/ln(10))+1, n=0..100); # Jaap Spies, Dec 11 2003

Table[Length[IntegerDigits[2^n]], {n, 0, 100}] (* T. D. Noe, Feb 11 2013 *)
IntegerLength[2^Range[0,80]] (* Harvey P. Dale, Jul 28 2017 *)

A034887(n)=n*log(2)\log(10)+1 \\ or: { a(n)=#digits(1<M. F. Hasler, Oct 08 2016

def a(n): return len(str(1 << n))
print([a(n) for n in range(73)]) # Michael S. Branicky, Dec 23 2022

a(n) = floor(n*log_10(2)) + 1. E.g., a(10)=4 because 2^10 = 1024 and floor(10*log_10(2)) + 1 = 3 + 1 = 4. - Jaap Spies, Dec 11 2003

a(n) = A055642(2^n) = A055642(A000079(n)).

A000461 Concatenate n n times.

Original entry on oeis.org

1, 22, 333, 4444, 55555, 666666, 7777777, 88888888, 999999999, 10101010101010101010, 1111111111111111111111, 121212121212121212121212, 13131313131313131313131313, 1414141414141414141414141414, 151515151515151515151515151515, 16161616161616161616161616161616

Offset: 1

John Radu (Suttones(AT)aol.com)

			From _Bruno Berselli_, Oct 05 2018: (Start)
.         1 * 9 = 09
.        22 * 9 = 198
.       333 * 9 = 2997
.      4444 * 9 = 39996
.     55555 * 9 = 499995
.    666666 * 9 = 5999994
.   7777777 * 9 = 69999993
.  88888888 * 9 = 799999992
. 999999999 * 9 = 8999999991
(End)

F. Smarandache, "Properties of the numbers", Univ. of Craiova Archives, 1975; Arizona State University Special Collections, Tempe, AZ.

Reinhard Zumkeller, Table of n, a(n) for n = 1..333
Eric Weisstein's World of Mathematics, Smarandache Sequences

Cf. A048376, A053422.

a000461 n = (read $ concat $ replicate n $ show n) :: Integer
-- Reinhard Zumkeller, Apr 26 2011

a:= n-> parse(cat(n$n)):
seq(a(n), n=1..20);  # Alois P. Heinz, Apr 26 2011

Table[Sum[(n)*10^(i*(Floor[Log[10, n]] + 1)), {i, 0, n - 1}], {n, 1, 30}] (* José de Jesús Camacho Medina, Dec 10 2014 *)
Table[FromDigits[Flatten[IntegerDigits/@Table[n,{n}]]],{n,15}] (* Harvey P. Dale, Mar 01 2015 *)
Table[FromDigits[PadRight[{},n IntegerLength[n],IntegerDigits[n]]],{n,15}] (* Harvey P. Dale, Jun 19 2016 *)

a(n) = eval(concat(apply(x->Str(x), vector(n, k, n)))); \\ Michel Marcus, Oct 05 2018; Feb 12 2023

def a(n): return int(str(n)*n)
print([a(n) for n in range(1, 17)]) # Michael S. Branicky, Jan 22 2021

a(n) = n*(10^(n*L(n))-1)/(10^L(n)-1) where L(n) = A004216(n)+1 = floor(log_10(10n)). - Henry Bottomley, Jun 01 2000
A055642(a(n)) = n * A055642(n). - Reinhard Zumkeller, Apr 26 2011
a(n) = Sum_{i=0..n-1} (n*10^(i*(floor(log(10, n)) + 1))). - José de Jesús Camacho Medina, Dec 10 2014

A050622 Numbers m that are divisible by 2^k, where k is the digit length of m.

Original entry on oeis.org

2, 4, 6, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 60, 64, 68, 72, 76, 80, 84, 88, 92, 96, 104, 112, 120, 128, 136, 144, 152, 160, 168, 176, 184, 192, 200, 208, 216, 224, 232, 240, 248, 256, 264, 272, 280, 288, 296, 304, 312, 320, 328, 336, 344, 352, 360

Offset: 1

Patrick De Geest, Jun 15 1999

The number of terms of length k is equal to (9*5^(k-1) - 1)/2. - Bernard Schott, Apr 06 2020

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

Cf. A035014, A050621, A055642.

seq(seq(j*2^k, j=(5^(k-1)+1)/2 .. 5^k-1),k=1..3); # Robert Israel, Apr 05 2020

Select[Range[360], IntegerQ[#/2^IntegerLength[#]] &] (* Jayanta Basu, May 25 2013 *)

isok(n) = n % (2^#Str(n)) == 0; \\ Michel Marcus, Sep 17 2015

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

A052018 Numbers k with the property that the sum of the digits of k is a substring of k.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 109, 119, 129, 139, 149, 159, 169, 179, 189, 199, 200, 300, 400, 500, 600, 700, 800, 900, 910, 911, 912, 913, 914, 915, 916, 917, 918, 919, 1000, 1009, 1018, 1027, 1036, 1045, 1054, 1063

Offset: 1

Patrick De Geest, Nov 15 1999

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A052019, A052020, A052021, A052022, A007953, A005349, A028834.
Cf. A175688. - Reinhard Zumkeller, Aug 15 2010
Cf. A055642, A004426.
Cf. A119246, A138166.

import Data.List (isInfixOf)
a052018 n = a052018_list !! (n-1)
a052018_list = filter f [0..] where
   f x = show (a007953 x) `isInfixOf` show x
-- Reinhard Zumkeller, Jun 18 2013

sdssQ[n_]:=Module[{idn=IntegerDigits[n],s,len},s=Total[idn];len= IntegerLength[ s]; MemberQ[Partition[idn,len,1],IntegerDigits[s]]]; Join[{0},Select[Range[1100],sdssQ]] (* Harvey P. Dale, Jan 02 2013 *)

loop = (str(n) for n in range(399))
print([int(n) for n in loop if str(sum(int(k) for k in n)) in n]) # Jonathan Frech, Jun 05 2017

A176995 Numbers that can be written as (m + sum of digits of m) for some m.

Original entry on oeis.org

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

Offset: 1

Reinhard Zumkeller, Aug 21 2011

The asymptotic density of this sequence is approximately 0.9022222 (Guaraldo, 1978). - Amiram Eldar, Nov 22 2020

			a(5) = 10 = 5 + (5);
a(87) = 100 = 86 + (8+6);
a(898) = 1000 = 977 + (9+7+7);
a(9017) = 10000 = 9968 + (9+9+6+8).

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).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
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.
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.
Eric Weisstein's World of Mathematics, Self Number.
Wikipedia, Self number.

Complement of A003052, range of A062028.

Cf. A062028, A007953, A055642, A230093.

a176995 n = a176995_list !! (n-1)
a176995_list = filter ((> 0) . a230093) [1..]
-- Reinhard Zumkeller, Oct 11 2013, Aug 21 2011

Select[Union[Table[n + Total[IntegerDigits[n]], {n, 77}]], # <= 77 &] (* Jayanta Basu, Jul 27 2013 *)

is_A003052(n)={for(i=1, min(n\2, 9*#digits(n)), sumdigits(n-i)==i && return); n} \\ from A003052
isok(n) = ! is_A003052(n) \\ Michel Marcus, Aug 20 2020

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

cp's OEIS Frontend

A004427 Arithmetic mean of digits of n (rounded up).

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A032810 Numbers using only digits 2 and 3.

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A047726 Number of different numbers that are formed by permuting digits of n.

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A160093 Number of digits in n, excluding any trailing zeros.

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A034887 Number of digits in 2^n.

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A000461 Concatenate n n times.

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