A162231 -id:A162231 - cp's OEIS frontend

A162232 Corresponding exponents for A162231.

1, 1, 1, 1, 1, 1, 1, 1, 4, 4, 2, 2, 3, 7, 7, 3, 4, 4, 4, 4, 3, 3, 3, 10, 10, 5, 5, 5, 13, 13, 8, 7, 7, 7, 7, 5, 5, 5, 5, 5, 16, 16, 8, 8, 6, 19, 19, 7, 7, 7, 10, 10, 10, 10, 7, 7, 8, 22, 22, 11, 11, 8, 25, 25, 9, 13, 13, 13, 13, 28, 28, 14, 14, 10, 10, 31, 31, 16, 16, 16, 16, 11, 12, 34, 34, 17

Offset: 1

Joseph Myers, Jun 28 2009

Joseph Myers, Table of n, a(n) for n=1..1130

Cf. A162231, A033840, A162233. In other bases: A162217 (base 3), A162220 (base 4), A162223 (base 5), A162226 (base 6), A162229 (base 7), 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

A033840 Smallest number > 1 equal to sum of n-th powers of its base-8 digits, or 0 if no such number exists (written in base 10).

Original entry on oeis.org

2, 20, 92, 16, 1056, 212419, 128, 13379, 40695508, 1024, 4196352, 13892162580, 8192, 268451840, 1002493281672, 65536, 17180000256, 4998382669357032, 524288, 1099512676352, 3418993385062038719, 4194304, 70368752566272

Offset: 1

David W. Wilson

Joseph Myers, Table of n, a(n) for n=1..170

Cf. A162231, A162232, A162233. In other bases: A033835 (base 3), A033836 (base 4), A033837 (base 5), A033838 (base 6), A033839 (base 7), A033841 (base 9), A003321 (base 10). - Joseph Myers, Jul 07 2009

Extended by 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

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

A162233 Greatest integer equal to the sum of the n-th powers of its base-8 digits (written in base 10).

Original entry on oeis.org

1, 7, 52, 433, 273, 25639, 212419, 2087646, 13606405, 40695508, 637339524, 6710775966, 32298119799, 497577637886, 268451841, 1002493281673, 66627168235658, 17180000257, 5190196317533094, 57033725899507007

Offset: 0

Joseph Myers, Jun 28 2009

Joseph Myers, Table of n, a(n) for n=0..168

Cf. A162231, A162232, A033840. In other bases: A162218 (base 3), A162221 (base 4), A162224 (base 5), A162227 (base 6), A162230 (base 7), A162236 (base 9), A046761 (base 10).

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

A162232 Corresponding exponents for A162231.

Views

Author

Keywords

Links

Crossrefs

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Extensions

A033840 Smallest number > 1 equal to sum of n-th powers of its base-8 digits, or 0 if no such number exists (written in base 10).

Views

Author

Keywords

Links

Crossrefs

Extensions

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.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

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.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

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.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

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.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

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.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A162233 Greatest integer equal to the sum of the n-th powers of its base-8 digits (written in base 10).

Views

Author

Keywords

Links

Crossrefs

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.

Views