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

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

Author

Gordon Hamilton, Dec 24 2004

Keywords

• base
• easy
• nonn

Comments

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

Examples

			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
		

Links

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

Crossrefs

Cf. A055642, A179239, A306360.

Programs

Magma

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

Mathematica

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

PARI

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

PARI

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

Python

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

Formula

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

Extensions

Name changed by Axel Harvey, Dec 26 2011

Edited by M. F. Hasler, Nov 17 2019