A162222 -id:A162222 - cp's OEIS frontend

A162223 Corresponding exponents for A162222.

1, 1, 1, 1, 1, 2, 2, 3, 3, 7, 4, 5, 4, 4, 6, 6, 6, 13, 8, 9, 9, 16, 14, 13, 15, 15, 14, 15, 15, 15, 16, 16, 16, 16, 16, 17, 17, 18, 18, 24, 20, 20, 20, 21, 21, 22, 22, 23, 24, 24, 25, 25, 25, 25, 25, 26, 26, 28, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 31, 31, 31, 32, 34, 34, 34, 34

Offset: 1

Joseph Myers, Jun 28 2009

Joseph Myers, Table of n, a(n) for n=1..1640

Cf. A162222, A033837, A162224. In other bases: A162217 (base 3), A162220 (base 4), A162226 (base 6), A162229 (base 7), A162232 (base 8), A162235 (base 9), A046074 (base 10).

A023052 Perfect Digital Invariants: numbers that are the sum of some fixed power of their digits.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 153, 370, 371, 407, 1634, 4150, 4151, 8208, 9474, 54748, 92727, 93084, 194979, 548834, 1741725, 4210818, 9800817, 9926315, 14459929, 24678050, 24678051, 88593477, 146511208, 472335975, 534494836, 912985153

Offset: 1

David W. Wilson

The old name was "Powerful numbers, definition (3)". Cf. A001694, A007532. - N. J. A. Sloane, Jan 16 2022.
Randle has suggested that these numbers be called "powerful", but this usually refers to a distinct property related to prime factorization, cf. A001694, A036966, A005934.
Numbers m such that m = Sum_{i=1..k} d(i)^s for some s, where d(1..k) are the decimal digits of m.
Superset of A005188 (Plusperfect, narcissistic or Armstrong numbers: s=k), A046197 (s=3), A052455 (s=4), A052464 (s=5), A124068 (s=6, 7), A124069 (s=8). - R. J. Mathar, Jun 15 2009, Jun 22 2009

			153 = 1^3 + 5^3 + 3^3, 4210818 = 4^7 + 2^7 + 1^7 + 0^7 + 8^7 + 1^7 + 8^7.

Jerome Raulin, Table of n, a(n) for n = 1..345 (terms 1..255 from Joseph Myers)
Encyclopaedia Britannica, Perfect digital invariant, article "Number patterns and curiosities" online since July 26, 1999, revised Aug 25, 2000.
Hans Havermann, Extended table of values for A023052 and A046074
Donald E. Knuth, The Art of Computer Programming, Volume 4, Pre-Fascicle 9B A Potpourri of Puzzles
J. Randle, Powerful numbers, Note 3208, Math. Gaz. 52 (1968), 383.
J. Randle, Powerful numbers, Note 3208, Math. Gaz. 52 (1968), 383. [Annotated scanned copy]
Eric Weisstein's World of Mathematics, Narcissistic Number
Index entries for sequences related to powerful numbers

Cf. A001694 (powerful numbers: p|n => p^2|n), A005934 (highly powerful numbers).
Cf. A003321, A007532, A014576, A046074, A046761, A053540, A161752.
Cf. A005188 (here the power must be equal to the number of digits).
In other bases: A162216 (base 3), A162219 (base 4), A162222 (base 5), A162225 (base 6), A162228 (base 7), A162231 (base 8), A162234 (base 9).

Select[Range[0, 10^5], Function[m, AnyTrue[Function[k, Total@ Map[Power[#, k] &, IntegerDigits@ m]] /@ Range@ 10, # == m &]]] (* Michael De Vlieger, Feb 08 2016, Version 10 *)

is(n)=if(n<10, return(1)); my(d=digits(n),m=vecmax(d)); if(m<2, return(0)); for(k=3,logint(n,m), if(sum(i=1,#d,d[i]^k)==n, return(1))); 0 \\ Charles R Greathouse IV, Feb 06 2017

select( is_A023052(n,b=10)={nn|| return(t==n))}, [0..10^5]) \\  M. F. Hasler, Nov 21 2019

Computed to 10^50 by G. N. Gusev (GGN(AT)rm.yaroslavl.ru)
Computed to 10^74 by Xiaoqing Tang
A-number typo corrected by R. J. Mathar, Jun 22 2009
Computed to 10^105 by Joseph Myers
Cross-references edited by Joseph Myers, Jun 28 2009
Edited by M. F. Hasler, Nov 21 2019

A033837 Smallest number > 1 equal to sum of n-th powers of its base-5 digits, or 0 if no such number exists (written in base 10).

Original entry on oeis.org

2, 13, 28, 289, 308, 4890, 257, 66562, 322217, 0, 0, 0, 16387, 268533762, 2204944815, 172449032, 34876823313, 207708636457, 0, 3315971951065, 8837942823632, 53027623387338, 422589088112942, 1129785254793

Offset: 1

David W. Wilson

Joseph Myers, Table of n, a(n) for n = 1..809

Cf. A162222, A162223, A162224. In other bases: A033835 (base 3), A033836 (base 4), A033838 (base 6), A033839 (base 7), A033840 (base 8), A033841 (base 9), A003321 (base 10). - Joseph Myers, Jul 07 2009

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

A162224 Greatest integer equal to the sum of the n-th powers of its base-5 digits (written in base 10).

Original entry on oeis.org

1, 4, 18, 118, 419, 308, 9113, 257, 66562, 1874374, 1, 1, 1, 338749352, 2415951874, 3264337734, 21690135624, 69236561678, 276040954847, 1, 4408512107192, 17665416905448, 70525678835838, 422589088112942, 1972301894061609

Offset: 0

Joseph Myers, Jun 28 2009

Joseph Myers, Table of n, a(n) for n=0..807

Cf. A162222, A162223, A033837. In other bases: A162218 (base 3), A162221 (base 4), A162227 (base 6), A162230 (base 7), A162233 (base 8), A162236 (base 9), A046761 (base 10).

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

cp's OEIS Frontend

A162223 Corresponding exponents for A162222.

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A023052 Perfect Digital Invariants: numbers that are the sum of some fixed power of their digits.

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A033837 Smallest number > 1 equal to sum of n-th powers of its base-5 digits, or 0 if no such number exists (written in base 10).

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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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A162224 Greatest integer equal to the sum of the n-th powers of its base-5 digits (written in base 10).

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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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