A010354 Base-8 Armstrong or narcissistic numbers (written in base 10).

1, 2, 3, 4, 5, 6, 7, 20, 52, 92, 133, 307, 432, 433, 16819, 17864, 17865, 24583, 25639, 212419, 906298, 906426, 938811, 1122179, 2087646, 3821955, 13606405, 40695508, 423056951, 637339524, 6710775966, 13892162580, 32298119799, 97095152738, 98250308556, 98317417420, 125586038802

Offset: 1

Author

N. J. A. Sloane

Keywords

• base
• fini
• full
• nonn

Comments

Like the other single-digit terms, zero would satisfy the definition (n = Sum_{i=1..k} d[i]^k when d[1..k] are the base 8 digits of n), but here only positive numbers are considered. - M. F. Hasler, Nov 20 2019

Examples

			From _M. F. Hasler_, Nov 20 2019: (Start)
20 = 24_8 (in base 8), and 2^2 + 4^2 = 20.
432 = 660_8, and 6^3 + 6^3 + 0^3 = 432; it's easy to see that 432 + 1 then also satisfies the equation, as for any term that is a multiple of 8. (End)
		

Links

• Joseph Myers, Table of n, a(n) for n = 1..62 (the full list of terms, from Winter)
• Eric Weisstein's World of Mathematics, Narcissistic Number
• D. T. Winter, Table of Armstrong Numbers

Crossrefs

Cf. A010351 (a(n) written in base 8).

In other bases: A010344 (base 4), A010346 (base 5), A010348 (base 6), A010350 (base 7), A010353 (base 9), A005188 (base 10), A161948 (base 11), A161949 (base 12), A161950 (base 13), A161951 (base 14), A161952 (base 15), A161953 (base 16).

Programs

PARI

select( {is_A010354(n)=n==vecsum([d^#n|d<-n=digits(n,8)])}, [0..10^6]) \\ This gives only terms < 10^6, for illustration of is_A010354(). - M. F. Hasler, Nov 20 2019

Python

from itertools import islice, combinations_with_replacement
def A010354_gen(): # generator of terms
    for k in range(1,30):
        a = tuple(i**k for i in range(8))
        yield from (x[0] for x in sorted(filter(lambda x:x[0] > 0 and tuple(int(d,8) for d in sorted(oct(x[0])[2:])) == x[1], \
                          ((sum(map(lambda y:a[y],b)),b) for b in combinations_with_replacement(range(8),k)))))
A010354_list = list(islice(A010354_gen(),20)) # Chai Wah Wu, Apr 20 2022

Extensions

Edited by Joseph Myers, Jun 28 2009