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

1,2

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

Links

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Wikipedia, Narcissistic number, as of Nov 18 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

Example

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

Mathematica

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

Prog

(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
(Magma) f:=func<n|&+[Intseq(n)[i]^#Intseq(n):i in [1..#Intseq(n)]]>; [f(n):n in [1..75]]; // Marius A. Burtea, Nov 18 2019

Crossrefs

Cf. A055642, A179239, A306360.

Sequence in context: A340270 A115026 A360075 * A135208 A259043 A156207

Adjacent sequences: A101334 A101335 A101336 * A101338 A101339 A101340

Keyword

base,easy,nonn

Author

Gordon Hamilton, Dec 24 2004

Extensions

Name changed by Axel Harvey, Dec 26 2011

Edited by M. F. Hasler, Nov 17 2019

Status

approved