A010346 -id:A010346 - cp's OEIS frontend

A161952 Base-15 Armstrong or narcissistic numbers (written in base 10).

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 113, 128, 2755, 3052, 5059, 49074, 49089, 386862, 413951, 517902, 15219156, 18605333, 38009273, 40082196, 40310423, 40868227, 47527794, 100128060, 100128061, 100128188, 104189152, 105464820

Offset: 1

Joseph Myers, Jun 22 2009

Whenever 15|a(n) (n = 32, 36, 40, 86, 100, 135, 143, 194, 197, 201), 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-15 digits of n), but this sequence only considers positive terms. - M. F. Hasler, Nov 22 2019

Joseph Myers, Table of n, a(n) for n = 1..202 (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
D. T. Winter, Table of Armstrong Numbers

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), A161953 (base 16).

Select[Range[10^7], # == Total[IntegerDigits[#, 15]^IntegerLength[#, 15]] &] (* Michael De Vlieger, Nov 04 2020 *)

select( is_A161952(n)={n==vecsum([d^#n|d<-n=digits(n,15)])}, [1..10^5]) \\ M. F. Hasler, Nov 22 2019

Terms sorted in increasing order by Pontus von Brömssen, Mar 03 2019

A162222 Base 5 perfect digital invariants (written in base 10): numbers equal to the sum of the k-th powers of their base-5 digits, for some k.

Original entry on oeis.org

0, 1, 2, 3, 4, 13, 18, 28, 118, 257, 289, 308, 353, 419, 4890, 4891, 9113, 16387, 66562, 322217, 1874374, 172449032, 268533762, 338749352, 2204944815, 2204944816, 2415951874, 3250054360, 3250054361, 3264337734, 4424304070, 4424304071

Offset: 1

Joseph Myers, Jun 28 2009

Whenever a(n) is a multiple of 5, then a(n+1) = a(n) + 1 is also a base 5 perfect digital invariant, with the same exponent k. - M. F. Hasler, Nov 21 2019

Joseph Myers, Table of n, a(n) for n=1..1640 (complete to 700 base 5 digits)

Cf. A162223 (corresponding exponents), A010346 (restriction to power = number of digits), A033837, A162224. In other bases: A162216 (base 3), A162219 (base 4), A162225 (base 6), A162228 (base 7), A162231 (base 8), A162234 (base 9), A023052 (base 10).

select( {is_A162222(n, b=5)=nn||return(t==n))}, [0..10^5]) \\ M. F. Hasler, Nov 21 2019

A010345 Base-5 Armstrong or narcissistic numbers, written in base 5.

Original entry on oeis.org

1, 2, 3, 4, 23, 33, 103, 433, 2124, 2403, 3134, 124030, 124031, 242423, 434434444, 1143204434402, 14421440424444

Offset: 1

N. J. A. Sloane

Also called Perfect Digital Invariant (PDI). When a(n) ends in 0, then a(n+1) = a(n) + 1 is also in the sequence, but in this base this only happens once. Zero would also satisfy the definition (n = Sum_{i=1..k} d[i]^k where d[1..k] are the base-5 digits of n), like the other single-digit terms. - M. F. Hasler, Nov 18 2019

The property of being an Armstrong number is an arithmetic property (like the number of divisors function) and is usually restricted to positive numbers. - N. J. A. Sloane, Nov 29 2019

Gordon L. Miller and Mary T. Whalen, Armstrong Numbers: 153 = 1^3 + 5^3 + 3^3, Fibonacci Quarterly, 30-3 (1992), 221-224. See Table 4 p. 223.
Eric Weisstein's World of Mathematics, Narcissistic Number
D. T. Winter, Table of Armstrong Numbers (latest backup on web.archive.org from Jan. 2010; page no longer available), published not later than Aug. 2003.

Cf. A010346 (a(n) written in base 10).

In other bases: A010343 (base 4), A010347 (base 6), A010349 (base 7), A010351 (base 8), A010352 (base 9), A005188 (base 10).

A010345=[fromdigits(digits(n,5))|n<-A010346] \\ Assumes the vector A010346 defined, see there for code. - M. F. Hasler, Nov 18 2019

Edited by Joseph Myers, Jun 28 2009

A357143 a(n) is sum of the base-5 digits of n each raised to the number of digits of n in base 5.

Original entry on oeis.org

1, 2, 3, 4, 1, 2, 5, 10, 17, 4, 5, 8, 13, 20, 9, 10, 13, 18, 25, 16, 17, 20, 25, 32, 1, 2, 9, 28, 65, 2, 3, 10, 29, 66, 9, 10, 17, 36, 73, 28, 29, 36, 55, 92, 65, 66, 73, 92, 129, 8, 9, 16, 35, 72, 9, 10, 17, 36, 73, 16, 17, 24, 43, 80, 35, 36, 43, 62, 99, 72, 73, 80, 99, 136, 27

Offset: 1

Francesco A. Catalanotti, Oct 26 2022

			For n = 13_10 = 23_5 (2 digits in base 5): a(13) = 2^2 + 3^2 = 13.
For n = 73_10 = 243_5 (3 digits in base 5): a(73) = 2^3 + 4^3 + 3^3 = 99.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A357954, A010346, A110592.

Cf. in base 10: A157714, A101337, A151544.

f:= proc(n) local L,d,i;
  L:= convert(n,base,5);
  d:= nops(L);
  add(L[i]^d,i=1..d)
end proc:
map(f,[$1..100]); # Robert Israel, Oct 26 2023

a[n_] := Total[IntegerDigits[n, 5]^IntegerLength[n, 5]]; Array[a, 100] (* Amiram Eldar, Oct 30 2022 *)

a(n) = my(d=digits(n, 5)); sum(k=1, #d, d[k]^#d); \\ Michel Marcus, Oct 29 2022

from sympy.ntheory.factor_ import digits
def A357143(n):
    t = len(s:=digits(n,5)[1:])
    return sum(d**t for d in s) # Chai Wah Wu, Oct 31 2022

a(n) = Sum_{i=1..A110592(n)} d(i)^A110592(n), where d(i) is the i-th digit of n in base 5.

Corrected and extended by Michel Marcus, Oct 29 2022

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

Original entry on oeis.org

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, 360672575087017687943, 264237343348909655564587, 267218514330351511200145

Offset: 1

Giovanni Corbelli, Mar 18 2022

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.

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

			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.)

Jinyuan Wang, C program
Jinyuan Wang, Table of base-20 Armstrong numbers
Eric Weisstein's World of Mathematics, Narcissistic Number

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).

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

isok(m) = my(d=digits(m, 20)); sum(k=1, #d, d[k]^#d) == m; \\ Michel Marcus, Mar 19 2022

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

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

a(31)-a(33) from Jinyuan Wang, May 05 2025

A357954 Integers k that are periodic points for some iterations of k->A357143(k).

Original entry on oeis.org

1, 2, 3, 4, 13, 18, 28, 118, 194, 289, 338, 353, 354, 419, 489, 528, 609, 769, 1269, 1299, 2081, 4890, 4891, 9113, 18575, 18702, 20759, 35084, 1874374, 338749352, 2415951874

Offset: 1

Francesco A. Catalanotti, Oct 22 2022

Given the function A357143(k), a number k is a term of the sequence if there exists a j such that A357143^j(k) = k, where j is the number of iterations applied.
The sequence is finite.
Proof: A357143(k) < k for all big enough k. g(k) = A110592(k)*4^A110592(k) is clearly an upper bound of A357143(k). Hence k > g(k) -> k > A357143(k), therefore every periodic point must be in an interval [s;t] such that for every k in [s;t] k <= g(k). Limit_{k->oo} g(k)/k = 0; now using the little-o definition we can show that there always exists a certain k_0 such that, for every k > k_0, k > g(k). The conclusion is that there must exist a finite number of intervals [s;t] and, consequently, a finite number of periodic points.
Every term k of the sequence is a periodic point (either a perfect digital invariant or a sociable digital invariant) for the function A357143(k).
The longest cycle needs 6 iterations to end: [489, 609, 769, 1269, 1299, 2081].

			k=9113 is a fixed point (perfect digital invariant) for the reiterated function A357143(k):
    9113_10 = 242423_5 (a 6-digit number in base 5);
    A357143(9113) = 2^6 + 4^6 + 2^6 + 4^6 + 2^6 + 3^6 = 9113.
k=18702 is a sociable digital invariant for the reiterated function A357143(k), requiring 2 iterations:
  1st iteration:
    18702_10 = 1044302_5 (a 7-digit number in base 5);
    A357143(18702) = 1^7 + 0^7 + 4^7 + 4^7 + 3^7 + 0^7 + 2^7 = 35084;
  2nd iteration:
    35084_10 = 2110314_5 (also a 7-digit number in base 5);
    A357143(35084) = 2^7 + 1^7 + 1^7 + 0^7 + 3^7 + 1^7 + 4^7 = 18702.

Wikipedia, Perfect digital invariant

Cf. A357143 , A010346 (fixed points), A110592 (exponents p(k)).

Cf. A157714 (base-10 sociable digital invariants), A101337 (A357143(k) base 10), A151544 (base-10 periodic points).

cp's OEIS Frontend

A161952 Base-15 Armstrong or narcissistic numbers (written in base 10).

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A162222 Base 5 perfect digital invariants (written in base 10): numbers equal to the sum of the k-th powers of their base-5 digits, for some k.

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A010345 Base-5 Armstrong or narcissistic numbers, written in base 5.

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A357143 a(n) is sum of the base-5 digits of n each raised to the number of digits of n in base 5.

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

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A357954 Integers k that are periodic points for some iterations of k->A357143(k).

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