A351374 Base-20 Armstrong or narcissistic numbers (written in base 10).

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 2413, 53808, 760400, 760401, 45661018, 62470211, 619939142, 14613048357, 1421043363262183, 48470736648305918, 514822672411130775

Offset

1,2

Comments

Written in base twenty the numbers are: 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F, G, H, I, J, 60D, 6EA8, 4F100, 4F101, E57CAI, JA8FAB, 9DEC7H2, B86BB0HH.

If a(28) exists, it is greater than 20^11.

Sequence is finite. Since k*19^k < 20^(k-1) for k >= 157, all terms must have less than 157 base-20 digits. 20*m is a term if and only if 20*m+1 is a term. - Chai Wah Wu, Apr 20 2022

Links

Table of n, a(n) for n=1..30.

Eric Weisstein's World of Mathematics, Narcissistic Number

Example

2413 is in the sequence because 2413 is 60D in base 20 (D stands for 13) and 6^3 + 0^3 + 13^3 = 2413. (The exponent 3 is the number of base-20 digits.)

Mathematica

Select[Range[10^6], # == Total[ IntegerDigits[#, 20]^IntegerLength[#, 20]] &]

Prog

(PARI) isok(m) = my(d=digits(m, 20)); sum(k=1, #d, d[k]^#d) == m; \\ Michel Marcus, Mar 19 2022
(Python)
from itertools import islice, combinations_with_replacement
from sympy.ntheory.factor_ import digits
def A351374_gen(): # generator of terms
    for k in range(1, 157):
        a = tuple(i**k for i in range(20))
        yield from (x[0] for x in sorted(filter(lambda x:x[0] > 0 and tuple(sorted(digits(x[0], 20)[1:])) == x[1], \
                          ((sum(map(lambda y:a[y], b)), b) for b in combinations_with_replacement(range(20), k)))))
A351374_list = list(islice(A351374_gen(), 20)) # Chai Wah Wu, Apr 20 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), A161953 (base 16).

Sequence in context: A229621 A077342 A242474 * A160544 A368728 A276568

Adjacent sequences: A351371 A351372 A351373 * A351375 A351376 A351377

Keyword

base,nonn,more,fini

Author

Giovanni Corbelli, Mar 18 2022

Extensions

a(28)-a(30) from Chai Wah Wu, Apr 20 2022

Status

approved