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A161953 Base-16 Armstrong or narcissistic numbers (written in base 10). 14
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
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
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
EXAMPLE
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.)
MATHEMATICA
Select[Range[10^7], # == Total[IntegerDigits[#, 16]^IntegerLength[#, 16]] &] (* Michael De Vlieger, Nov 04 2020 *)
PROG
(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
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).
Sequence in context: A043320 A044917 A246337 * A187829 A356974 A105427
KEYWORD
base,fini,full,nonn
AUTHOR
Joseph Myers, Jun 22 2009
EXTENSIONS
Terms sorted in increasing order by Pontus von Brömssen, Mar 03 2019
STATUS
approved

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Last modified March 28 10:31 EDT 2024. Contains 371240 sequences. (Running on oeis4.)