A010354 -id:A010354 - cp's OEIS frontend

A005188 Armstrong (or pluperfect, or Plus Perfect, or narcissistic) numbers: m-digit positive numbers equal to sum of the m-th powers of their digits.

1, 2, 3, 4, 5, 6, 7, 8, 9, 153, 370, 371, 407, 1634, 8208, 9474, 54748, 92727, 93084, 548834, 1741725, 4210818, 9800817, 9926315, 24678050, 24678051, 88593477, 146511208, 472335975, 534494836, 912985153, 4679307774, 32164049650, 32164049651

Offset: 1

N. J. A. Sloane, Robert G. Wilson v

A finite sequence, the 88th and last term being 115132219018763992565095597973971522401.
Let k = d_1 d_2 ... d_n in base 10; then k is in the sequence iff k = Sum_{i=1..n} d_i^n.
These are the fixed points in the "Recurring Digital Invariant Variant" described in A151543.
a(15) = A229381(3) = 8208 is the "Simpsons' narcissistic number".
If a(n) is a multiple of 10, then a(n+1) = a(n) + 1, and if a(n) == 1 (mod 10) then a(n-1) = a(n) - 1 except for n = 1, cf. Examples. - M. F. Hasler, Oct 18 2018
Named after Michael Frederick Armstrong (1941-2020), who used these numbers in his computing class at the University of Rochester in the mid 1960's. - Amiram Eldar, Mar 09 2024

			153 = 1^3 + 5^3 + 3^3,
8208 = 8^4 + 2^4 + 0^4 + 8^4,
4210818 = 4^7 + 2^7 + 1^7 + 0^7 + 8^7 + 1^7 + 8^7.
The eight terms 370, 24678050, 32164049650, 4338281769391370, 3706907995955475988644380, 19008174136254279995012734740, 186709961001538790100634132976990 and 115132219018763992565095597973971522400 end in a digit zero, therefore their successor a(n) + 1 is the next term a(n+1). This also yields the last term of the sequence. The initial a(1) = 1 is the only term ending in a digit 1 not preceded by a(n) - 1. - _M. F. Hasler_, Oct 18 2018

Jean-Marie De Koninck, Ces nombres qui nous fascinent, Entry 88, pp. 30-31, Ellipses, Paris 2008.
Lionel E. Deimel, Jr. and Michael T. Jones, Finding Pluperfect Digital Invariants: Techniques, Results and Observations, J. Rec. Math., 14 (1981), 87-108.
Jean-Pierre Lamoitier, Fifty Basic Exercises. SYBEX Inc., 1981.
Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 68.
Alfred S. Posamentier, Numbers: Their Tales, Types, and Treasures, Prometheus Books, 2015, pp. 242-244.
Joe Roberts, The Lure of the Integers, The Mathematical Association of America, 1992, page 36.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..88 (the full list of terms, from Winter)
Anonymous, Narcissistic number.
Michael F. Armstrong, A Brief Introduction to Armstrong Numbers.
Pat Ballew, The Cubic Attractiveness of 153, Pat's Blog, May 30, 2023.
Hans J. de Jong, Letter to N. J. A. Sloane, Mar 8 1988.
Lionel E. Deimel, Armstrong Numbers.
Lionel E. Deimel, Mystery Solved!, Lionel Deimel's Web Log, May 5, 2010.
Lionel E. Deimel, Narcissistic Numbers.
Martin Gardner & N. J. A. Sloane, Correspondence, 1973-74.
Shyam Sunder Gupta, Digital Invariants and Narcissistic Numbers, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 21, 513-526.
Harvey Heinz, Narcissistic Numbers (backup from March 2018 on web/archive.org: page no longer available), Sep. 1998, last updated in Sep. 2010.
History of Science and Mathematics StackExchange, Armstrong numbers - who is or was Armstrong?, 2021.
L. H. & W. Lopez, PlanetMath.Org, Armstrong number (latest backup on web.archive.org of ArmstrongNumber.html from 2012), published by L.H. not later than July 2007.
Gordon L. Miller and Mary T. Whalen, Armstrong Numbers: 153 = 1^3 + 5^3 + 3^3, Fibonacci Quarterly, 30-3 (1992), 221-224.
Tomas Antonio Mendes Oliveira e Silva (tos(AT)ci.ua.pt), Loneliness of the Factorions, gave the full sequence in a posting (Article 42889) to sci.math on May 09 1994.
Walter Schneider, Perfect Digital Invariants: Pluperfect Digital Invariants(PPDIs)
B. Shader, Armstrong number.
Eric Weisstein's World of Mathematics, Narcissistic Number.
Wikipedia, Narcissistic number.
Robert G. Wilson v, Letter to N. J. A. Sloane, Jan 23 1989.
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. A001694, A007532, A005934, A003321, A014576, A046074.
Similar to but different from A023052.
Cf. A151543.
Cf. A010343 to A010354 (bases 4 to 9). - R. J. Mathar, Jun 28 2009

filter:= proc(k) local d;
d:= 1 + ilog10(k);
add(s^d, s=convert(k,base,10)) = k
end proc:
select(filter, [$1..10^6]); # Robert Israel, Jan 02 2015

f[n_] := Plus @@ (IntegerDigits[n]^Floor[ Log[10, n] + 1]); Select[ Range[10^7], f[ # ] == # &] (* Robert G. Wilson v, May 04 2005 *)
Select[Range[10^7],#==Total[IntegerDigits[#]^IntegerLength[#]]&] (* Harvey P. Dale, Sep 30 2011 *)

is(n)=my(v=digits(n));sum(i=1,#v,v[i]^#v)==n \\ Charles R Greathouse IV, Nov 20 2012

select( is_A005188(n)={n==vecsum([d^#n|d<-n=digits(n)])}, [0..9999]) \\ M. F. Hasler, Nov 18 2019

from itertools import combinations_with_replacement
A005188_list = []
for k in range(1,10):
    a = [i**k for i in range(10)]
    for b in combinations_with_replacement(range(10),k):
        x = sum(map(lambda y:a[y],b))
        if x > 0 and tuple(int(d) for d in sorted(str(x))) == b:
            A005188_list.append(x)
A005188_list = sorted(A005188_list) # Chai Wah Wu, Aug 25 2015

32164049651 from Amit Munje (amit.munje(AT)gmail.com), Oct 07 2006
In order to agree with the Definition, first comment modified by Jonathan Sondow, Jan 02 2015
Comment in name moved to comment section and links edited by M. F. Hasler, Oct 18 2018
"Positive" added to definition by N. J. A. Sloane, Nov 18 2019

A010346 Base-5 Armstrong or narcissistic numbers (written in base 10).

Original entry on oeis.org

1, 2, 3, 4, 13, 18, 28, 118, 289, 353, 419, 4890, 4891, 9113, 1874374, 338749352, 2415951874

Offset: 1

N. J. A. Sloane

Zero would also satisfy the definition as the other single-digit terms, but here only positive numbers are considered. - M. F. Hasler, Nov 20 2019

Eric Weisstein's World of Mathematics, Narcissistic Number
D. T. Winter, Table of Armstrong Numbers

Cf. A010345 (a(n) written in base 5).

In other bases: A010344 (base 4), 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).

A010346=select( is_A010346(n)={n==vecsum([d^#n|d<-n=digits(n,5)])}, [0..9999]) \\ This yields only terms < 10^4 (i.e., all but the last 3 terms), for illustration of is_A010346(). In older versions of PARI, use {n==sum(i=1,#n=digits(n,5),n[i]^#n)}. - M. F. Hasler, Nov 20 2019

Edited by Joseph Myers, Jun 28 2009

A010344 Base-4 Armstrong or narcissistic numbers (written in base 10).

Original entry on oeis.org

1, 2, 3, 28, 29, 35, 43, 55, 62, 83, 243

Offset: 1

N. J. A. Sloane

Eric Weisstein's World of Mathematics, Narcissistic Number
D. T. Winter, Table of Armstrong Numbers

Cf. A010343 (a(n) written in base 4).

In other bases: 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).

A010344=select( n->n==vecsum([d^#n|d<-n=digits(n,4)]), [0..333]) \\ M. F. Hasler, Nov 18 2019

Edited by Joseph Myers, Jun 28 2009

A010348 Base-6 Armstrong or narcissistic numbers (written in base 10).

Original entry on oeis.org

1, 2, 3, 4, 5, 99, 190, 2292, 2293, 2324, 3432, 3433, 6197, 36140, 269458, 391907, 10067135, 2510142206, 2511720147, 3866632806, 3866632807, 3930544834, 4953134588, 5018649129, 6170640875, 124246559501, 4595333541803, 5341093125744, 5341093125745, 19418246235419

Offset: 1

N. J. A. Sloane

From M. F. Hasler, Nov 20 2019: (Start)
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 digits of n), but here only positive numbers are considered.
Terms a(n+1) = a(n) + 1 (n = 8, 11, 20, 28) correspond to solutions a(n) ending in a digit 0 in base 6, in which case a(n) + 1 also is a solution. (End)

Joseph Myers, Table of n, a(n) for n = 1..30 (the full list of terms, from Winter)
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. A010347 (a(n) written in base 6).

In other bases: A010344 (base 4), A010346 (base 5), 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( {is_A010348(n)=n==vecsum([d^#n|d<-n=digits(n,6)])}, [0..4e5\1]) \\ Note: this yields only terms < 10^6, for illustration of is_A010348(). - M. F. Hasler, Nov 20 2019

Edited by Joseph Myers, Jun 28 2009

A010350 Base-7 Armstrong or narcissistic numbers (written in base 10).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 10, 25, 32, 45, 133, 134, 152, 250, 3190, 3222, 3612, 3613, 4183, 9286, 35411, 191334, 193393, 376889, 535069, 794376, 8094840, 10883814, 16219922, 20496270, 32469576, 34403018, 416002778, 416352977, 420197083, 725781499, 1500022495, 15705029375, 15705029376, 28700208851

Offset: 1

N. J. A. Sloane

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 digits of n), but here only positive numbers are considered. - M. F. Hasler, Nov 20 2019

Joseph Myers, Table of n, a(n) for n = 1..59 (the full list of terms, from Winter)
Eric Weisstein's World of Mathematics, Narcissistic Number
D. T. Winter, Table of Armstrong Numbers

Cf. A010349 (a(n) written in base 7).

In other bases: A010344 (base 4), A010346 (base 5), A010348 (base 6), 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( {is_A010350(n)=n==vecsum([d^#n|d<-n=digits(n,7)])}, [0..10^6]) \\ This yields only terms < 10^6, for illustration of is_A010350(). - M. F. Hasler, Nov 20 2019

Edited by Joseph Myers, Jun 28 2009

A010353 Base-9 Armstrong or narcissistic numbers (written in base 10).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 41, 50, 126, 127, 468, 469, 1824, 8052, 8295, 9857, 1198372, 3357009, 3357010, 6287267, 156608073, 156608074, 403584750, 403584751, 586638974, 3302332571, 42256814922, 42256814923, 114842637961, 155896317510, 552468844242, 552468844243, 647871937482, 686031429775

Offset: 1

N. J. A. Sloane

From M. F. Hasler, Nov 20 2019: (Start)
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 9 digits of n), but here only positive numbers are considered.
Terms a(n+1) = a(n) + 1 (n = 11, 13, 20, 23, 25, 29, 33, 48, 51, 57) correspond to solutions a(n) that are multiples of 9, in which case a(n) + 1 is also a solution. (End)

			126 = 150_9 (= 1*9^2 + 5*9^1 + 0*9^0) = 1^3 + 5^3 + 0^3. It is easy to see that 126 + 1 then also satisfies this relation, as for all other terms that are multiples of 9. - _M. F. Hasler_, Nov 20 2019

Joseph Myers, Table of n, a(n) for n = 1..58 (the full list of terms, from Winter)
René-Louis Clerc, Perfect r-narcissistic numbers in any base, hal-04376934, 2024.
Eric Weisstein's World of Mathematics, Narcissistic Number
D. T. Winter, Table of Armstrong Numbers

Cf. A010352 (a(n) written in base 9).

In other bases: A010344 (base 4), A010346 (base 5), A010348 (base 6), A010350 (base 7), A010354 (base 8), A005188 (base 10), A161948 (base 11), A161949 (base 12), A161950 (base 13), A161951 (base 14), A161952 (base 15), A161953 (base 16).

Select[Range[9^7], # == Total[IntegerDigits[#, 9]^IntegerLength[#, 9]] &] (* Michael De Vlieger, Jan 17 2024 *)

select( {is_A010353(n)=n==vecsum([d^#n|d<-n=digits(n,9)])}, [0..10^4]) \\ This gives only terms < 10^6, for illustration of is_A010353(). - M. F. Hasler, Nov 20 2019

Edited by Joseph Myers, Jun 28 2009

A161949 Base-12 Armstrong or narcissistic numbers (written in base 10).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 29, 125, 811, 944, 1539, 28733, 193084, 887690, 2536330, 6884751, 17116683, 5145662993, 25022977605, 39989277598, 294245206529, 301149802206, 394317605931, 429649124722, 446779986586

Offset: 1

Joseph Myers, Jun 22 2009

Joseph Myers, Table of n, a(n) for n = 1..87 (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), A161950 (base 13), A161951 (base 14), A161952 (base 15), A161953 (base 16).

Select[Range[10^7], # == Total[IntegerDigits[#, 12]^IntegerLength[#, 12]] &] (* Michael De Vlieger, Nov 04 2020 *)

A161953 Base-16 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, 342, 371, 520, 584, 645, 1189, 1456, 1457, 1547, 1611, 2240, 2241, 2458, 2729, 2755, 3240, 3689, 3744, 3745, 47314, 79225, 177922, 177954, 368764, 369788, 786656, 786657, 787680, 787681, 811239, 812263, 819424, 819425, 820448, 820449, 909360

Offset: 1

Joseph Myers, Jun 22 2009

Whenever 16|a(n) (n = 22, 26, 33, 41, 43, 47, 49, 51, 53, 61, 116, 149, 157, 196, 198, 204, 206, 243, 247), 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-16 digits of n), but this sequence only considers positive terms. - M. F. Hasler, Nov 22 2019

			645 is in the sequence because 645 is 285 in hexadecimal and 2^3 + 8^3 + 5^3 = 645. (The exponent 3 is the number of hexadecimal digits.)

Joseph Myers, Table of n, a(n) for n=1..293 (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), A161952 (base 15).

Select[Range[10^7], # == Total[IntegerDigits[#, 16]^IntegerLength[#, 16]] &] (* Michael De Vlieger, Nov 04 2020 *)

isok(n) = {my(b=16, d=digits(n, b), e=#d); sum(k=1, #d, d[k]^e) == n;} \\ Michel Marcus, Feb 25 2019

select( is_A161953(n)={n==vecsum([d^#n|d<-n=digits(n,16)])}, [1..10^5]) \\ M. F. Hasler, Nov 22 2019

from itertools import islice, combinations_with_replacement
def A161953_gen(): # generator of terms
    for k in range(1,74):
        a = tuple(i**k for i in range(16))
        yield from (x[0] for x in sorted(filter(lambda x:x[0] > 0 and tuple(int(d,16) for d in sorted(hex(x[0])[2:])) == x[1], \
                          ((sum(map(lambda y:a[y],b)),b) for b in combinations_with_replacement(range(16),k)))))
A161953_list = list(islice(A161953_gen(),30)) # Chai Wah Wu, Apr 21 2022

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

A161948 Base-11 Armstrong or narcissistic numbers (written in base 10).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 61, 72, 126, 370, 855, 1161, 1216, 1280, 10657, 16841, 16842, 17864, 17865, 36949, 36950, 63684, 66324, 71217, 90120, 99594, 99595, 141424, 157383, 1165098, 1165099, 5611015, 11959539, 46478562, 203821954, 210315331, 397800208, 826098079, 1308772162, 1399714480

Offset: 1

Joseph Myers, Jun 22 2009

From M. F. Hasler, Nov 20 2019: (Start)
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 11 digits of n), but here only positive numbers are considered.
Terms a(n+1) = a(n) + 1 (n = 20, 22, 24, 30, 34, 56, 67, 57, 195, ...) correspond to solutions a(n) that are multiples of 11, in which case a(n) + 1 is also a solution. (End)

			16841 = 11720_11 (= 1*11^4 + 1*11^3 + 7*11^2 + 2*11^1 + 0*11^0) = 1^5 + 1^5 + 7^5 + 2^5 + 0^5. It's easy to see that 16841 + 1 then also satisfies this relation, as for all terms that are multiples of 11. - _M. F. Hasler_, Nov 20 2019

Joseph Myers, Table of n, a(n) for n = 1..134 (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), A161949 (base 12), A161950 (base 13), A161951 (base 14), A161952 (base 15), A161953 (base 16).

Select[Range[10^7], # == Total[IntegerDigits[#, 11]^IntegerLength[#, 11]] &] (* Michael De Vlieger, Nov 04 2020 *)

select( {is_A161948(n)=n==vecsum([d^#n|d<-n=digits(n,11)])}, [0..10^5]) \\ This gives only terms < 10^5, for illustration of is_A161948(). - M. F. Hasler, Nov 20 2019

A161950 Base-13 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, 17, 45, 85, 98, 136, 160, 793, 794, 854, 1968, 8194, 62481, 167544, 167545, 294094, 320375, 323612, 325471, 325713, 350131, 365914, 2412003, 4861352, 21710514, 43757311, 43757312, 46299414, 51798568, 52994053

Offset: 1

Joseph Myers, Jun 22 2009

Joseph Myers, Table of n, a(n) for n = 1..201 (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

Select[Range[10^7], # == Total[IntegerDigits[#, 13]^IntegerLength[#, 13]] &] (* Michael De Vlieger, Nov 04 2020 *)

cp's OEIS Frontend

A005188 Armstrong (or pluperfect, or Plus Perfect, or narcissistic) numbers: m-digit positive numbers equal to sum of the m-th powers of their digits.

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A010346 Base-5 Armstrong or narcissistic numbers (written in base 10).

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A010344 Base-4 Armstrong or narcissistic numbers (written in base 10).

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A010348 Base-6 Armstrong or narcissistic numbers (written in base 10).

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A010350 Base-7 Armstrong or narcissistic numbers (written in base 10).

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A010353 Base-9 Armstrong or narcissistic numbers (written in base 10).

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A161949 Base-12 Armstrong or narcissistic numbers (written in base 10).

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A161953 Base-16 Armstrong or narcissistic numbers (written in base 10).

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