A351374 - OEIS

%I #65 Apr 21 2022 09:16:07

%S 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,2413,53808,760400,

%T 760401,45661018,62470211,619939142,14613048357,1421043363262183,

%U 48470736648305918,514822672411130775

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

%C 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.

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

%C 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

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/NarcissisticNumber.html">Narcissistic Number</a>

%e 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.)

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

%o (PARI) isok(m) = my(d=digits(m, 20)); sum(k=1, #d, d[k]^#d) == m; \\ _Michel Marcus_, Mar 19 2022

%o (Python)

%o from itertools import islice, combinations_with_replacement

%o from sympy.ntheory.factor_ import digits

%o def A351374_gen(): # generator of terms

%o     for k in range(1,157):

%o         a = tuple(i**k for i in range(20))

%o         yield from (x[0] for x in sorted(filter(lambda x:x[0] > 0 and tuple(sorted(digits(x[0],20)[1:])) == x[1], \

%o                           ((sum(map(lambda y:a[y],b)),b) for b in combinations_with_replacement(range(20),k)))))

%o A351374_list = list(islice(A351374_gen(),20)) # _Chai Wah Wu_, Apr 20 2022

%Y 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).

%K base,nonn,more,fini

%O 1,2

%A _Giovanni Corbelli_, Mar 18 2022

%E a(28)-a(30) from _Chai Wah Wu_, Apr 20 2022