A161953 Base-16 Armstrong or narcissistic numbers (written in base 10).

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 342, 371, 520, 584, 645, 1189, 1456, 1457, 1547, 1611, 2240, 2241, 2458, 2729, 2755, 3240, 3689, 3744, 3745, 47314, 79225, 177922, 177954, 368764, 369788, 786656, 786657, 787680, 787681, 811239, 812263, 819424, 819425, 820448, 820449, 909360

Offset: 1

Author

Joseph Myers, Jun 22 2009

Keywords

• base
• fini
• full
• nonn

Comments

Whenever 16|a(n) (n = 22, 26, 33, 41, 43, 47, 49, 51, 53, 61, 116, 149, 157, 196, 198, 204, 206, 243, 247), then a(n+1) = a(n) + 1. Zero also satisfies the definition (n = Sum_{i=1..k} d[i]^k where d[1..k] are the base-16 digits of n), but this sequence only considers positive terms. - M. F. Hasler, Nov 22 2019

Examples

			645 is in the sequence because 645 is 285 in hexadecimal and 2^3 + 8^3 + 5^3 = 645. (The exponent 3 is the number of hexadecimal digits.)
		

Links

• Joseph Myers, Table of n, a(n) for n=1..293 (the full list of terms, from Winter)
• Henk Koppelaar and Peyman Nasehpour, On Hardy's Apology Numbers, arXiv:2008.08187 [math.NT], 2020.
• Eric Weisstein's World of Mathematics, Narcissistic Number
• D. T. Winter, Table of Armstrong Numbers

Crossrefs

In other bases: A010344 (base 4), A010346 (base 5), A010348 (base 6), A010350 (base 7), A010354 (base 8), A010353 (base 9), A005188 (base 10), A161948 (base 11), A161949 (base 12), A161950 (base 13), A161951 (base 14), A161952 (base 15).

Programs

Mathematica

Select[Range[10^7], # == Total[IntegerDigits[#, 16]^IntegerLength[#, 16]] &] (* Michael De Vlieger, Nov 04 2020 *)

PARI

isok(n) = {my(b=16, d=digits(n, b), e=#d); sum(k=1, #d, d[k]^e) == n;} \\ Michel Marcus, Feb 25 2019

PARI

select( is_A161953(n)={n==vecsum([d^#n|d<-n=digits(n,16)])}, [1..10^5]) \\ M. F. Hasler, Nov 22 2019

Python

from itertools import islice, combinations_with_replacement
def A161953_gen(): # generator of terms
    for k in range(1,74):
        a = tuple(i**k for i in range(16))
        yield from (x[0] for x in sorted(filter(lambda x:x[0] > 0 and tuple(int(d,16) for d in sorted(hex(x[0])[2:])) == x[1], \
                          ((sum(map(lambda y:a[y],b)),b) for b in combinations_with_replacement(range(16),k)))))
A161953_list = list(islice(A161953_gen(),30)) # Chai Wah Wu, Apr 21 2022

Extensions

Terms sorted in increasing order by Pontus von Brömssen, Mar 03 2019