A023052 -id:A023052 - cp's OEIS frontend

A014576 Smallest n-digit narcissistic (or Armstrong) number: smallest n-digit number equal to sum of n-th powers of its digits (or 0 if no such number exists).

1, 0, 153, 1634, 54748, 548834, 1741725, 24678050, 146511208, 4679307774, 32164049650, 0, 0, 28116440335967, 0, 4338281769391370, 21897142587612075, 0, 1517841543307505039, 63105425988599693916, 128468643043731391252, 0

Offset: 1

N. J. A. Sloane

M. Gardner, The Magic Numbers of Dr Matrix. Prometheus, Buffalo, NY, 1985, p. 249.
C. A. Pickover, Keys to Infinity. New York: W. H. Freeman, pp. 169-170, 1995.

T. D. Noe, Table of n, a(n) for n=1..39 (complete sequence)
Harvey Heinz, Narcissistic Numbers
Mike Keith, Wild Narcissistic Numbers (Copy as of May 2008 on web.archive.org - page does not exist any more.)
Eric Weisstein's World of Mathematics, Narcissistic Number.

Cf. A001694, A007532, A005934, A005188, A003321, A023052, A046074.

(* This program is not suitable for more than 10 terms *) a[n_] := For[k = 10^(n-1), True, k++, If[k > 10^n - 1, Return[0], If[k == Total[ IntegerDigits[k]^IntegerLength[k] ], Return[k] ] ] ]; Table[ Print[an = a[n]]; an, {n, 1, 10}] (* Jean-François Alcover, Oct 15 2013 *)

Terms and links added by Patrick De Geest, Oct 1998

Broken links fixed by M. F. Hasler, Feb 12 2013

A162216 Base-3 perfect digital invariants (written in base 10): numbers equal to the sum of the k-th powers of their base-3 digits, for some k.

Original entry on oeis.org

0, 1, 2, 5, 8, 17, 33, 34, 65, 66, 67, 131, 258, 259, 386, 512, 513, 514, 1026, 1027, 2049, 2050, 3075, 3076, 4100, 16388, 16389, 16390, 57345, 57346, 65538, 65539, 196610, 262149, 262150, 458754, 458755, 786438, 786439, 1048581, 1048582, 1310724

Offset: 1

Joseph Myers, Jun 28 2009

Whenever 3|a(n), then a(n+1) = a(n) + 1 (for the same k). The first 6 terms are exactly all the base-3 narcissistic numbers (where k = number of base-3 digits). For these numbers in other bases b = 4, ..., 16 see A010344 - A161953. - M. F. Hasler, Nov 18 2019

Joseph Myers, Table of n, a(n) for n = 1..6130 (complete to 2000 base-3 digits)

Cf. A162217 (corresponding exponents), A033835, A162218. In other bases: A162219 (base 4), A162222 (base 5), A162225 (base 6), A162228 (base 7), A162231 (base 8), A162234 (base 9), A023052 (base 10).

select( is_A162216(n,b=3)={nn||return(s==n))}, [0..10^5]) \\ M. F. Hasler, Nov 21 2019

A162219 Base 4 perfect digital invariants (written in base 10): numbers equal to the sum of the k-th powers of their base-4 digits, for some k.

Original entry on oeis.org

0, 1, 2, 3, 8, 9, 28, 29, 32, 33, 35, 43, 55, 62, 83, 128, 129, 243, 512, 513, 922, 2048, 2049, 2316, 2317, 2444, 2445, 2571, 2699, 7330, 8192, 8193, 13124, 13125, 20710, 21222, 32768, 32769, 40392, 40393, 131072, 131073, 524288, 524289, 1075174

Offset: 1

Joseph Myers, Jun 28 2009

Whenever a(n) is a multiple of 4, then a(n+1) = a(n) + 1 is also a base 4 perfect digital invariant, with the same exponent k. - M. F. Hasler, Nov 21 2019

Joseph Myers, Table of n, a(n) for n=1..6778 (complete to 1500 base 4 digits)

Cf. A162220 (corresponding exponents), A010344 (restriction to power = number of digits), A033836, A162221. In other bases: A162216 (base 3), A162222 (base 5), A162225 (base 6), A162228 (base 7), A162231 (base 8), A162234 (base 9), A023052 (base 10).

select( {is_A162219(n, b=4)=nn||return(s==n))}, [0..10^5]) \\ M. F. Hasler, Nov 21 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

A162225 Base-6 perfect digital invariants (written in base 10): numbers equal to the sum of the k-th powers of their base-6 digits, for some k.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 99, 190, 251, 308, 2292, 2293, 2324, 3432, 3433, 6197, 36140, 269458, 391907, 10067135, 1428423394, 2510142206, 2511720147, 3866632806, 3866632807, 3930544834, 4953134588, 5018649129, 6170640875, 32693825124, 32693825125

Offset: 1

Joseph Myers, Jun 28 2009

Whenever a(n) is a multiple of 6, then a(n+1) = a(n) + 1 is also a base-6 perfect digital invariant, with the same exponent k. - M. F. Hasler, Nov 21 2019

Joseph Myers, Table of n, a(n) for n = 1..764 (complete to 350 base-6 digits)

Cf. A162226 (corresponding exponents), A010348 (restriction to power = number of digits), A033838, A162227. In other bases: A162216 (base 3), A162219 (base 4), A162222 (base 5), A162228 (base 7), A162231 (base 8), A162234 (base 9), A023052 (base 10).

select( {is_A162225(n, b=6)=if(n1 && forstep(p=logint(n,t), logint(n, vecsum(b)), -1, (t=vecsum([d^p|d<-b]))>n|| return(t==n)))}, [0..10^5]) \\ M. F. Hasler, Nov 21 2019

A162228 Base 7 perfect digital invariants (written in base 10): numbers equal to the sum of the k-th powers of their base-7 digits, for some k.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 9, 10, 16, 25, 32, 45, 65, 133, 134, 152, 250, 1542, 3190, 3222, 3612, 3613, 4183, 9286, 35411, 37271, 72865, 191334, 193393, 376889, 535069, 794376, 1110699, 2236488, 3021897, 4431562, 8094840, 9885773, 10883814, 16219922

Offset: 1

Joseph Myers, Jun 28 2009

Whenever a(n) is a multiple of 7, then a(n+1) = a(n) + 1 is also a base 7 perfect digital invariant, with the same exponent k. - M. F. Hasler, Nov 21 2019

Joseph Myers, Table of n, a(n) for n=1..868 (complete to 200 base 7 digits)

Cf. A162229 (corresponding exponents), A010350 (restriction to power = number of digits), A033839, A162230. In other bases: A162216 (base 3), A162219 (base 4), A162222 (base 5), A162225 (base 6), A162231 (base 8), A162234 (base 9), A023052 (base 10).

select( {is_A162228(n, b=7)=nn||return(s==n))}, [0..10^5]) \\ M. F. Hasler, Nov 21 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

A162234 Base 9 perfect digital invariants (written in base 10): numbers equal to the sum of the k-th powers of their base-9 digits, for some k.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 27, 28, 41, 50, 126, 127, 243, 244, 353, 468, 469, 1052, 1824, 2187, 2188, 8052, 8295, 9857, 19683, 19684, 36804, 65538, 65539, 177147, 177148, 1198372, 1594323, 1594324, 3357009, 3357010, 5300099, 6287267, 10097892

Offset: 1

Joseph Myers, Jun 28 2009

Whenever a(n) is a multiple of 9, then a(n+1) = a(n) + 1 is also a base 9 perfect digital invariant, with the same exponent k. - M. F. Hasler, Nov 21 2019

Joseph Myers, Table of n, a(n) for n=1..506 (complete to 120 base 9 digits)

Cf. A162235 (corresponding exponents), A010353 (restriction to power = number of digits), A033841, A162236. In other bases: A162216 (base 3), A162219 (base 4), A162222 (base 5), A162225 (base 6), A162228 (base 7), A162231 (base 8), A023052 (base 10).

select( {is_A162234(n, b=9)=nn|| return(t==n))}, [0..10^5]) \\ M. F. Hasler, Nov 21 2019

A052464 Fixed points for operation of repeatedly replacing a number with the sum of the fifth power of its digits.

Original entry on oeis.org

0, 1, 4150, 4151, 54748, 92727, 93084, 194979

Offset: 1

Henry Bottomley, Mar 15 2000

Equivalently, numbers equal to the sum of 5th powers of their decimal digits. Since this sum is <= 9^5*d for a d-digit number n >= 10^(d-1), there cannot be such a number with more than 6 digits. - M. F. Hasler, Apr 12 2015

			a(2) = 4150 since 4^5 + 1^5 + 5^5 + 0^5 = 1024 + 1 + 3125 + 0 = 4150.

G. K. Patil, Ramanujan's Life And His Contributions In The Field Of Mathematics, International Journal of Scientific Research and Engineering Studies (IJSRES), 1(6) (2014), ISSN: 2349-8862.

Cf. A023052, A046074, A046197, A052455, A124068, A124069, A226970, A003321.

for(n=0,10^6,A055014(n)==n&&print1(n",")) \\ M. F. Hasler, Apr 12 2015

A053540 a(n) = n*9^(n-1).

Original entry on oeis.org

1, 18, 243, 2916, 32805, 354294, 3720087, 38263752, 387420489, 3874204890, 38354628411, 376572715308, 3671583974253, 35586121596606, 343151886824415, 3294258113514384, 31501343210481297, 300189270593998242, 2851798070642983299, 27017034353459841780

Offset: 1

Barry E. Williams, Jan 15 2000

Vincenzo Librandi, Table of n, a(n) for n = 1..300
Frank Ellermann, Illustration of binomial transforms.
Index entries for linear recurrences with constant coefficients, signature (18,-81).

Related to computing A023052.

Cf. A001787, A053464, A053469.

List([1..20], n-> n*9^(n-1)); # G. C. Greubel, May 16 2019

[n*9^(n-1): n in [1..20]]; // Vincenzo Librandi, Jun 11 2011

f[n_]:=n*9^(n-1); f[Range[40]] (* Vladimir Joseph Stephan Orlovsky, Feb 09 2011*)

a(n)=n*9^(n-1) \\ Charles R Greathouse IV, Oct 07 2015

[n*9^(n-1) for n in (1..20)] # G. C. Greubel, May 16 2019

From Colin Barker, Oct 17 2012: (Start)
a(n) = 18*a(n-1) - 81*a(n-2).
G.f.: x/(1-9*x)^2. (End)
E.g.f.: x*exp(9*x). - G. C. Greubel, May 16 2019
From Amiram Eldar, Oct 28 2020: (Start)
Sum_{n>=1} 1/a(n) = 9*log(9/8).
Sum_{n>=1} (-1)^(n+1)/a(n) = 9*log(10/9). (End)

More terms from Larry Reeves (larryr(AT)acm.org), May 29 2001

Edited by N. J. A. Sloane at the suggestion of Reinhard Zumkeller, Sep 16 2007

cp's OEIS Frontend

A014576 Smallest n-digit narcissistic (or Armstrong) number: smallest n-digit number equal to sum of n-th powers of its digits (or 0 if no such number exists).

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A162216 Base-3 perfect digital invariants (written in base 10): numbers equal to the sum of the k-th powers of their base-3 digits, for some k.

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A162219 Base 4 perfect digital invariants (written in base 10): numbers equal to the sum of the k-th powers of their base-4 digits, for some k.

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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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A162225 Base-6 perfect digital invariants (written in base 10): numbers equal to the sum of the k-th powers of their base-6 digits, for some k.

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A162228 Base 7 perfect digital invariants (written in base 10): numbers equal to the sum of the k-th powers of their base-7 digits, for some k.

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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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A162234 Base 9 perfect digital invariants (written in base 10): numbers equal to the sum of the k-th powers of their base-9 digits, for some k.

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A052464 Fixed points for operation of repeatedly replacing a number with the sum of the fifth power of its digits.

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