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

1,2

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

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

Example

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)

Prog

(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

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).

Sequence in context: A200333 A024644 A297182 * A070759 A282138 A106275

Adjacent sequences: A010351 A010352 A010353 * A010355 A010356 A010357

Keyword

base,fini,full,nonn

Author

N. J. A. Sloane

Extensions

Edited by Joseph Myers, Jun 28 2009

Status

approved