A306360 -id:A306360 - cp's OEIS frontend

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

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

A101337 Sum of (each digit of n raised to the power (number of digits in n)).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 5, 10, 17, 26, 37, 50, 65, 82, 4, 5, 8, 13, 20, 29, 40, 53, 68, 85, 9, 10, 13, 18, 25, 34, 45, 58, 73, 90, 16, 17, 20, 25, 32, 41, 52, 65, 80, 97, 25, 26, 29, 34, 41, 50, 61, 74, 89, 106, 36, 37, 40, 45, 52, 61, 72, 85, 100, 117, 49, 50, 53, 58, 65

Offset: 1

Gordon Hamilton, Dec 24 2004

Sometimes referred to as "narcissistic function" (in base 10). Fixed points are the narcissistic (or Armstrong, or plus perfect) numbers A005188. - M. F. Hasler, Nov 17 2019

			a(75) = 7^2 + 5^2 = 74 and a(705) = 7^3 + 0^3 + 5^3 = 468.
a(1.02e59 - 1) = 102429587095122578993551250282047487264694110769657513064859 ~ 1.024e59 is an example of n close to the limit beyond which a(n) < n for all n. - _M. F. Hasler_, Nov 17 2019

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Wikipedia, Narcissistic number, as of Nov 18 2019.

Cf. A055642, A179239, A306360.

f:=func; [f(n):n in [1..75]]; // Marius A. Burtea, Nov 18 2019

Array[Total[IntegerDigits[#]^IntegerLength[#]]&,80] (* Harvey P. Dale, Aug 27 2011 *)

a(n)=my(d=digits(n)); sum(i=1,#d, d[i]^#d) \\ Charles R Greathouse IV, Aug 10 2017

apply( A101337(n)=vecsum([d^#n|d<-n=digits(n)]), [0..99]) \\ M. F. Hasler, Nov 17 2019

def A101337(n):
    s = str(n)
    l = len(s)
    return sum(int(d)**l for d in s) # Chai Wah Wu, Feb 26 2019

a(n) <= A055642(n)*9^A055642(n) with equality for all n = 10^k - 1. Write n = 10^x to get a(n) < n when 1+log_10(x+1) < (x+1)(1-log_10(9)) <=> x > 59.85. It appears that a(n) < n already for all n > 1.02*10^59. - M. F. Hasler, Nov 17 2019

Name changed by Axel Harvey, Dec 26 2011

Edited by M. F. Hasler, Nov 17 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

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

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.

A334601 Positive integers m such that sum of cubes of the digits of m, t=A055012(m), is a multiple of m (m/A055012(m) is an integer >= 1).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 24, 27, 37, 48, 153, 370, 371, 407, 459

Offset: 1

Zak Seidov, May 07 2020 and May 12 2020

Corresponding values of t: 1, 8, 27, 64, 125, 216, 343, 512, 729, 72, 351, 370, 576, 153, 370, 371, 407, 918 (first 9 terms are all cubes).
Corresponding values of t/m: 1, 4, 9, 16, 25, 36, 49, 64, 81, 3, 13, 10, 12, 1, 1, 1, 1, 2 (first 9 terms are all squares).
The subsequence of numbers m such that sum of cubes of its digits is equal to m is A046197 \ {0}. - Bernard Schott, May 11 2020

			m = 459, t = 4^3 + 5^3 + 9^3 = 918, t/m = 2.

Cf. A005188, A046197, A046459, A055012, A055642, A101337, A306354, A306360, A306361, A329291.

Select[Range[500], Divisible[Plus @@ (IntegerDigits[#]^3), #] &] (* Amiram Eldar, May 11 2020 *)

isok(m) = my(d=digits(m)); sum(k=1, #d, d[k]^3) % m == 0; \\ Michel Marcus, May 14 2020

cp's OEIS Frontend

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

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A101337 Sum of (each digit of n raised to the power (number of digits in n)).

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

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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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A334601 Positive integers m such that sum of cubes of the digits of m, t=A055012(m), is a multiple of m (m/A055012(m) is an integer >= 1).

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