A101337 -id:A101337 - cp's OEIS frontend

A306360 Numbers k such that A101337(k)/k is an integer.

1, 2, 3, 4, 5, 6, 7, 8, 9, 153, 370, 371, 407, 459, 1634, 8208, 9474, 13598, 48495, 54748, 92727, 93084, 119564, 174961, 306979, 548834, 1741725, 3194922, 4210818, 9800817, 9926315, 12720569, 24678050, 24678051, 88593477, 144688641, 146511208

Offset: 1

Ctibor O. Zizka, Feb 10 2019

A005188 is a subsequence of this sequence.

Sequence is finite. In particular, a(n) < 10^60. If k >= 10^60, then A101337(k) < k. - Chai Wah Wu, Feb 26 2019

			For k = 1, (1^1)/1 = 1;
for k = 459, (4^3 + 5^3 + 9^3) / 459 = 2.

Giovanni Resta, Table of n, a(n) for n = 1..109 (full sequence; first 56 terms from Chai Wah Wu)
Barry Fagin, Idempotent Factorizations of Square-free Integers, Preprints (2019).

Cf. A005188, A101337, A306354, A306361.

Select[Range[10^6], IntegerQ[Total[IntegerDigits[#]^IntegerLength[#]]/#] &] (* Michael De Vlieger, Aug 01 2019 *)

isok(n) = frac(A101337(n)/n) == 0; \\ Michel Marcus, Feb 11 2019

select( is(n)=!(A101337(n)%n), [0..999]) \\ M. F. Hasler, Nov 17 2019

A306360_list, k = [], 1
while k < 10**9:
    s = str(k)
    l, c = len(s), 0
    for i in range(l):
        c = (c + int(s[i])**l) % k
    if c == 0:
        A306360_list.append(k)
    k += 1 # Chai Wah Wu, Feb 26 2019

a(22)-a(37) from Daniel Suteu, Feb 10 2019

A306354 a(n) = gcd(n, A101337(n)).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 2, 1, 4, 1, 2, 1, 4, 1, 3, 1, 1, 3, 1, 1, 9, 1, 1, 3, 8, 1, 2, 1, 4, 1, 2, 1, 16, 1, 25, 1, 1, 1, 1, 5, 1, 1, 1, 1, 12, 1, 2, 9, 4, 1, 6, 1, 4, 3, 7, 1, 1, 1, 1, 1, 1, 7, 1, 1, 16, 1, 2, 1, 4, 1, 2, 1, 8, 1, 9

Offset: 1

Ctibor O. Zizka, Feb 09 2019

A101337(n) / n = r, r an integer, gives A306360. A101337(n) / n = 1 gives A005188. n / A101337(n) = s, s an integer, gives A306361. The motivation for this sequence was the question as to which numbers n have the property A101337(n) / n = r and the property n / A101337(n) = s?

			For n = 24, a(24) = gcd(24, 2*2 + 4*4) = gcd(24,20) = 4, thus a(24) = 4;
for n = 153, a(153) = gcd(153, 1*1*1 + 5*5*5 + 3*3*3) = gcd(153,153) = 153, thus a(153) = 153.

Cf. A005188, A066750, A101337.

Array[GCD[#1, Total[#2^Length[#2]]] & @@ {#, IntegerDigits@ #} &, 90] (* Michael De Vlieger, Feb 09 2019 *)

a(n) = my(d=digits(n)); gcd(n, sum(i=1, #d, d[i]^#d)); \\ Michel Marcus, Feb 12 2019

from math import gcd
def A306354(n): return gcd(n,sum(int(d)**len(str(n)) for d in str(n))) # Chai Wah Wu, Jan 26 2022

A306361 Numbers k divisible by A101337(k) (narcissistic function).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 20, 50, 100, 110, 111, 153, 200, 221, 370, 371, 407, 500, 702, 1000, 1010, 1011, 1020, 1100, 1101, 1110, 1121, 1122, 1634, 2000, 2322, 4104, 5000, 8208, 9474, 10000, 10010, 10011, 10100, 10101, 10110, 11000, 11001, 11010, 11022, 11100, 11122, 11220, 12012, 12110, 12210, 12320, 14550

Offset: 1

Ctibor O. Zizka, Feb 10 2019

A005188 is a subsequence of this sequence.
Numbers in A007088 with either 3 or 9 ones are terms of this sequence. - Chai Wah Wu, Feb 26 2019
For all N in A007088 we have A101337(N) = A007953(N) = number of digits '1'; whenever this equals 2^k*5^m (k, m >= 0) and N ends in max(k,m) '0's, then N is also in this sequence. - M. F. Hasler, Nov 18 2019

			For k = 20, 20 / (2^2 + 0^2) = 5;
for k = 221, 221 / (2^3 + 2^3 + 1^3) = 13.

Cf. A005188, A007088, A101337, A306354, A306360.

isok(n) = frac(n/A101337(n)) == 0; \\ Michel Marcus, Feb 11 2019

select( is_A306361=t->!(t%A101337(t)), [0..9999]) \\ M. F. Hasler, Nov 18 2019

A329291 Ratio A101337(m)/m for m = A306360(n), numbers dividing the value of their narcissistic function.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 7, 2, 1, 1, 1, 5, 4, 4, 1, 1, 3, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 1, 3, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 4, 1, 1, 3, 2, 1, 4, 1, 1, 1, 4, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

M. F. Hasler, Nov 17 2019

A101337 (sum of digits raised to power A055642(n) = #digits(n)) is sometimes called the narcissistic function for base 10.

This sequence has the same finite number of elements as A306360.

Cf. A101337, A306360, A329292 (least index i with a(i) = n = 1, 2, 3...).

PARI
```
A329291=[A101337(m)/m|m<-A306360]
```

a(n) = A101337(A306360(n))/A306360(n).

a(108)-a(109) from Giovanni Resta, Nov 18 2019

A329292 Least number m > 0 such that A101337(m)/m = n, or 0 if no such m exists.

Original entry on oeis.org

1, 459, 3194922, 174961, 119564, 0, 13598

Offset: 1

M. F. Hasler, Nov 17 2019

Subsequence of A306360, therefore also finite.

Cf. A329291, A101337, A306360.

{A329292=vector(vecmax(A329291),k, iferr(for(i=1,#A329291, A329291[i]==k&&error(k=A306360[i])),E,k))} \\ If A329291 is not complete, there may be erroneous 0's

a(n) = min { m in A306360 | A101337(m)/m = n }.

a(6)-a(7) from Chai Wah Wu, Nov 18 2019 using b-file of A306360

A115026 Limiting value of n under iteration of "sum of the digits raised to the power of the number of digits of n" (A101337).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 5, 1, 370, 370, 370, 370, 370, 1, 4, 5, 8, 1, 4, 370, 370, 370, 1, 370, 9, 1, 1, 370, 370, 370, 370, 370, 370, 370, 370, 370, 4, 370, 1, 370, 370, 370, 370, 1, 370, 370, 370, 370, 370, 370, 370, 370, 370, 160, 370, 370, 370, 370, 370, 370

Offset: 1

Sergio Pimentel, Feb 24 2006

Iterate A101337 starting at n until reaching a constant value (like 370) or a cycle (like 160, 217, 352, 160, ...). In the latter case, a(n) takes the smallest value in the cycle (e.g., a(59) = 160). Since k*9^k < 10^k for all k > 34, each number n is guaranteed to yield a smaller number a(n) if n > 10^34, so every number reaches a constant or a cycle under this sequence.

Conjecture: no term is greater than 370. - Harvey P. Dale, Jun 08 2022

			a(89)=370 since:
89 (2 digits): 8^2 + 9^2 = 145,
145 (3 digits): 1^3 + 4^3 + 5^3 = 190,
190 (3 digits): 1^3 + 9^3 + 0^3 = 730,
730 (3 digits): 7^3 + 3^3 + 0^3 = 370,
370 (3 digits): 3^3 + 7^3 + 0^3 = 370, etc.
So a(89) = 370 since 370 is a fixed point of A101337.

Cf. A101337.

Table[Min[FindTransientRepeat[NestList[Total[IntegerDigits[#]^IntegerLength[#]]&,n,20],3][[2]]],{n,70}] (* Harvey P. Dale, Jun 08 2022 *)

A329659 Largest number m > 0 such that A101337(m)/m = n, or 0 if no such m exists.

Original entry on oeis.org

115132219018763992565095597973971522401, 13384899942524140745922870, 18935132531699388, 2098649524599800996, 119564, 0, 13598

Offset: 1

Chai Wah Wu, Nov 18 2019

Cf. A101337, A329292, A306360.

A007088 The binary numbers (or binary words, or binary vectors, or binary expansion of n): numbers written in base 2.

Original entry on oeis.org

0, 1, 10, 11, 100, 101, 110, 111, 1000, 1001, 1010, 1011, 1100, 1101, 1110, 1111, 10000, 10001, 10010, 10011, 10100, 10101, 10110, 10111, 11000, 11001, 11010, 11011, 11100, 11101, 11110, 11111, 100000, 100001, 100010, 100011, 100100, 100101, 100110, 100111

Offset: 0

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

List of binary numbers. (This comment is to assist people searching for that particular phrase. - N. J. A. Sloane, Apr 08 2016)
Or, numbers that are sums of distinct powers of 10.
Or, numbers having only digits 0 and 1 in their decimal representation.
Complement of A136399; A064770(a(n)) = a(n). - Reinhard Zumkeller, Dec 30 2007
From Rick L. Shepherd, Jun 25 2009: (Start)
Nonnegative integers with no decimal digit > 1.
Thus nonnegative integers n in base 10 such that kn can be calculated by normal addition (i.e., n + n + ... + n, with k n's (but not necessarily k + k + ... + k, with n k's)) or multiplication without requiring any carry operations for 0 <= k <= 9. (End)
For n > 1: A257773(a(n)) = 10, numbers that are Belgian-k for k=0..9. - Reinhard Zumkeller, May 08 2015
For any integer n>=0, find the binary representation and then interpret as decimal representation giving a(n). - Michael Somos, Nov 15 2015
N is in this sequence iff A007953(N) = A101337(N). A028897 is a left inverse. - M. F. Hasler, Nov 18 2019
For n > 0, numbers whose largest decimal digit is 1. - Stefano Spezia, Nov 15 2023

			a(6)=110 because (1/2)*((1-(-1)^6)*10^0 + (1-(-1)^3)*10^1 + (1-(-1)^1)*10^2) = 10 + 100.
G.f. = x + 10*x^2 + 11*x^3 + 100*x^4 + 101*x^5 + 110*x^6 + 111*x^7 + 1000*x^8 + ...
.
  000    The numbers < 2^n can be regarded as vectors with
  001    a fixed length n if padded with zeros on the left
  010    side. This represents the n-fold Cartesian product
  011    over the set {0, 1}. In the example on the left,
  100    n = 3. (See also the second Python program.)
  101    Binary vectors in this format can also be seen as a
  110    representation of the subsets of a set with n elements.
  111    - _Peter Luschny_, Jan 22 2024

John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 21.
Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §2.8 Binary, Octal, Hexadecimal, p. 64.
Manfred R. Schroeder, "Fractals, Chaos, Power Laws", W. H. Freeman, 1991, p. 383.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane, Table of n, a(n) for n = 0..32768 (first 8192 terms from Franklin T. Adams-Watters)
Heinz Gumin, Herrn von Leibniz' Rechnung mit Null und Eins, Siemens AG, 3. Auflage 1979 -- contains facsimiles of Leibniz's papers from 1679 and 1703.
Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, Identities and periodic oscillations of divide-and-conquer recurrences splitting at half, arXiv:2210.10968 [cs.DS], 2022, p. 45.
G. W. Leibniz, Explication de l'arithmétique binaire, qui se sert des seuls caractères 0 & 1; avec des remarques sur son utilité, et sur ce qu'elle donne le sens des anciennes figures chinoises de Fohy, Mémoires de l'Académie Royale des Sciences, 1703, pp. 85-89; reprinted in Gumin (1979).
N. J. A. Sloane, Table of a(n) for n = 0..1048576 (A large file).
Robert G. Wilson v, Letter to N. J. A. Sloane, Sep. 1992.
Index entries for sequences related to Most Wanted Primes video.
Index entries for 10-automatic sequences.
Index entries for sequences related to binary expansion of n.

The basic sequences concerning the binary expansion of n are this one, A000120 (Hammingweight: sum of bits), A000788 (partial sums of A000120), A000069 (A000120 is odd), A001969 (A000120 is even), A023416 (number of bits 0), A059015 (partial sums). Bisections A099820 and A099821.
Cf. A028897 (convert binary to decimal).
Cf. A000042, A007089-A007095, A000695, A005836, A033042-A033052, A159918, A004290, A169965, A169966, A169967, A169964, A204093, A204094, A204095, A097256, A257773, A257770.
Cf. A000290, A001737, A007953, A030308, A054055, A064770, A101337, A136399.

a007088 0 = 0
a007088 n = 10 * a007088 n' + m where (n',m) = divMod n 2
-- Reinhard Zumkeller, Jan 10 2012

A007088 := n-> convert(n, binary): seq(A007088(n), n=0..50); # R. J. Mathar, Aug 11 2009

Table[ FromDigits[ IntegerDigits[n, 2]], {n, 0, 39}]
Table[Sum[ (Floor[( Mod[f/2 ^n, 2])])*(10^n) , {n, 0, Floor[Log[2, f]]}], {f, 1, 100}] (* José de Jesús Camacho Medina, Jul 24 2014 *)
FromDigits/@Tuples[{1,0},6]//Sort (* Harvey P. Dale, Aug 10 2017 *)

{a(n) = subst( Pol( binary(n)), x, 10)}; /* Michael Somos, Jun 07 2002 */

{a(n) = if( n<=0, 0, n%2 + 10*a(n\2))}; /* Michael Somos, Jun 07 2002 */

a(n)=fromdigits(binary(n),10) \\ Charles R Greathouse IV, Apr 08 2015

def a(n): return int(bin(n)[2:])
print([a(n) for n in range(40)]) # Michael S. Branicky, Jan 10 2021

from itertools import product
n = 4
for p in product([0, 1], repeat=n): print(''.join(str(x) for x in p))
# Peter Luschny, Jan 22 2024

a(n) = Sum_{i=0..m} d(i)*10^i, where Sum_{i=0..m} d(i)*2^i is the base 2 representation of n.
a(n) = (1/2)*Sum_{i>=0} (1-(-1)^floor(n/2^i))*10^i. - Benoit Cloitre, Nov 20 2001
a(n) = A097256(n)/9.
a(2n) = 10*a(n), a(2n+1) = a(2n)+1.
G.f.: 1/(1-x) * Sum_{k>=0} 10^k * x^(2^k)/(1+x^(2^k)) - for sequence as decimal integers. - Franklin T. Adams-Watters, Jun 16 2006
a(A000290(n)) = A001737(n). - Reinhard Zumkeller, Apr 25 2009
a(n) = Sum_{k>=0} A030308(n,k)*10^k. - Philippe Deléham, Oct 19 2011
For n > 0: A054055(a(n)) = 1. - Reinhard Zumkeller, Apr 25 2012
a(n) = Sum_{k=0..floor(log_2(n))} floor((Mod(n/2^k, 2)))*(10^k). - José de Jesús Camacho Medina, Jul 24 2014

A113019 (Number of digits of n) raised to the power of (the digital root of n).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 2, 4, 8, 16, 32, 64, 128, 256, 512, 2, 4, 8, 16, 32, 64, 128, 256, 512, 2, 4, 8, 16, 32, 64, 128, 256, 512, 2, 4, 8, 16, 32, 64, 128, 256, 512, 2, 4, 8, 16, 32, 64, 128, 256, 512, 2, 4, 8, 16, 32

Offset: 0

Alexandre Wajnberg, Jan 03 2006

n=1 and 32 are fixed points. Are there any others?

First occurrence of k: 1,10,100,11,10000,100000,1000000,12,101,1000000000, ..., . - Robert G. Wilson v

			a(0) = 1^0 = 1.
a(9) = 1^9 = 1.
a(10) = 2^(1+0) = 2.
a(89) = 2^(8+9=17=>1+7) = 2^8 = 256.

Nathaniel Johnston, Table of n, a(n) for n = 0..10000

Cf. A101337.

A113019 := proc(n) if(n=0)then return 1:fi: return length(n)^(((n-1) mod 9) + 1): end: seq(A113019(n),n=0..100); # Nathaniel Johnston, May 04 2011

f[n_] := If[n == 0, 1, Floor[ Log[10, 10n]]^(Mod[n - 1, 9] + 1)]; Table[ f[n], {n, 0, 73}] (* Robert G. Wilson v, Jan 04 2006 *)

apply( A113019(n)=(logint(n+!n,10)+1)^((n-1)%9+1), [0..99]) \\ M. F. Hasler, Nov 17 2019

a(ijk...) [m digits ijk...] = m^(i+j+k+..[one digit])

a(n)=A055642(n)^A010888(n). - Robert G. Wilson v

A113009 {Sum of the digits of n} raised to the power {number of digits of n}.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 36, 49, 64, 81, 100, 121, 144, 169, 196

Offset: 0

Alexandre Wajnberg, Jan 03 2006

Fixed points are: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 81, 5832. Are there any others?

Fixed points include: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 81, 512, 2401. There are no other fixed points less than 10^1000. - Chai Wah Wu, Feb 28 2019

			a(9)=9^1=9
a(19)=(1+9)^2=100
a(101)=(1+0+1)^3=8

Cf. A101337.

Join[{0},Table[Total[IntegerDigits[n]]^IntegerLength[n],{n,100}]] (* Harvey P. Dale, Nov 09 2014 *)

def A113009(n):
    return sum(int(d) for d in str(n))**len(str(n)) # Chai Wah Wu, Feb 28 2019

a(ijk...)[m digits ijk...]=(i+j+k)^m

cp's OEIS Frontend

A306360 Numbers k such that A101337(k)/k is an integer.

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A306354 a(n) = gcd(n, A101337(n)).

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A306361 Numbers k divisible by A101337(k) (narcissistic function).

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A329291 Ratio A101337(m)/m for m = A306360(n), numbers dividing the value of their narcissistic function.

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A329292 Least number m > 0 such that A101337(m)/m = n, or 0 if no such m exists.

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A115026 Limiting value of n under iteration of "sum of the digits raised to the power of the number of digits of n" (A101337).

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A329659 Largest number m > 0 such that A101337(m)/m = n, or 0 if no such m exists.

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A007088 The binary numbers (or binary words, or binary vectors, or binary expansion of n): numbers written in base 2.

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