A010354 -id:A010354 - cp's OEIS frontend

A161951 Base-14 Armstrong or narcissistic numbers (written in base 10).

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 244, 793, 282007, 10362564, 1445712420, 29546248981, 164159496751, 342515735622, 359057049845, 216210334578515, 324075236456868, 338527182572746, 338609726265795, 382789516519507, 435198066019184, 526088332647250

Offset: 1

Joseph Myers, Jun 22 2009

Whenever 14|a(n) (n = 36, 46, 75, 77), 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-14 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..103 (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), A161952 (base 15), A161953 (base 16).

Select[Range[2 * 10^7], # == Total[IntegerDigits[#, 14]^IntegerLength[#, 14]] &] (* Michael De Vlieger, Nov 04 2020 *)

select( is_A161951(n)={n==vecsum([d^#n|d<-n=digits(n,14)])}, [1..10^6\3]) \\ M. F. Hasler, Nov 22 2019

A161952 Base-15 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, 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

A162231 Base 8 perfect digital invariants (written in base 10): numbers equal to the sum of the k-th powers of their base-8 digits, for some k.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 16, 17, 20, 52, 92, 128, 129, 133, 256, 257, 272, 273, 307, 432, 433, 1024, 1025, 1056, 1057, 2323, 8192, 8193, 13379, 16384, 16385, 16512, 16513, 16819, 17864, 17865, 24583, 25639, 65536, 65537, 65792, 65793, 212419, 524288, 524289

Offset: 1

Joseph Myers, Jun 28 2009

Whenever a(n) is a multiple of 8, then a(n+1) = a(n) + 1 is also a base 8 perfect digital invariant, with the same exponent k. - M. F. Hasler, Nov 21 2019

Joseph Myers, Table of n, a(n) for n=1..1130 (complete to 160 base 8 digits)

Cf. A162232 (corresponding exponents), A010354 (restriction to power = number of digits), A033840, A162233. In other bases: A162216 (base 3), A162219 (base 4), A162222 (base 5), A162225 (base 6), A162228 (base 7), A162234 (base 9), A023052 (base 10).

select( is_A162231(n,b=8)={nn|| return(t==n))}, [0..10^5]) \\ M. F. Hasler, Nov 21 2019

A010351 Base-8 Armstrong or narcissistic numbers, written in base 8.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 24, 64, 134, 205, 463, 660, 661, 40663, 42710, 42711, 60007, 62047, 636703, 3352072, 3352272, 3451473, 4217603, 7755336, 16450603, 63717005, 233173324, 3115653067, 4577203604, 61777450236, 147402312024

Offset: 1

N. J. A. Sloane

Whenever a term ends in 0, then a(n+1) = a(n) + 1 is also a term. 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 18 2019

			432 = 660_8 (= 6*8^2 + 6*8^1 + 0*8^0), and 6^3 + 6^3 + 0^3 = 432, therefore 660 is in the sequence. It's easy to see that 432 + 1 then also satisfies the equation, as for any term that is a multiple of 8. - _M. F. Hasler_, Nov 21 2019

Joseph Myers, Table of n, a(n) for n = 1..62 (the full list of terms, from Winter)
Gordon L. Miller and Mary T. Whalen, Armstrong Numbers: 153 = 1^3 + 5^3 + 3^3, Fibonacci Quarterly, 30-3 (1992), 221-224.
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. A010354 (a(n) written in base 10).

In other bases: A010343 (base 4), A010345 (base 5), A010347 (base 6), A010349 (base 7), A010352 (base 9), A005188 (base 10).

[fromdigits(digits(n,8))|n<-A010354] \\ M. F. Hasler, Nov 18 2019

Edited by Joseph Myers, Jun 28 2009

A180015 List of Armstrong (narcissistic) numbers in base 8 that are prime (written in base 10).

Original entry on oeis.org

2, 3, 5, 7, 307, 433, 25639, 212419, 1122179

Offset: 1

Robin James Kerrison (rjk1994(AT)gmail.com), Aug 06 2010

			The term 25639 is equal to 62047 in base 8. 62047 has 5 digits, and 6^5 + 2^5 + 0^5 + 4^5 + 7^5 = 62047 (in base 8) = 25639 (in base 10).

Cf. A010354

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

cp's OEIS Frontend

A161951 Base-14 Armstrong or narcissistic numbers (written in base 10).

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

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A162231 Base 8 perfect digital invariants (written in base 10): numbers equal to the sum of the k-th powers of their base-8 digits, for some k.

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A010351 Base-8 Armstrong or narcissistic numbers, written in base 8.

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A180015 List of Armstrong (narcissistic) numbers in base 8 that are prime (written in base 10).

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Author

Keywords

Examples

Crossrefs

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

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Author

Keywords

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Examples

Links

Crossrefs

Programs

Mathematica

PARI

Python

Extensions