cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 10 results.

A162217 Corresponding exponents for A162216.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 5, 5, 5, 6, 6, 6, 8, 8, 7, 7, 9, 9, 10, 10, 9, 9, 10, 10, 10, 12, 14, 14, 13, 13, 14, 14, 15, 16, 16, 16, 16, 18, 18, 18, 18, 18, 18, 18, 18, 19, 19, 20, 22, 22, 24, 24, 26, 26, 25, 26, 26, 27, 28, 29, 29, 29, 29, 30, 30, 31, 31, 32, 32, 32, 32, 33, 33, 33, 33, 33
Offset: 1

Views

Author

Joseph Myers, Jun 28 2009

Keywords

Links

Crossrefs

Cf. A162216, A033835, A162218. In other bases: A162220 (base 4), A162223 (base 5), 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

Views

Author

David W. Wilson

Keywords

Comments

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

Examples

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

Links

Crossrefs

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).

Programs

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

Extensions

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

A033835 Smallest number > 1 equal to sum of n-th powers of its base-3 digits, or 0 if no such number exists (written in base 10).

Original entry on oeis.org

2, 5, 17, 0, 33, 66, 386, 258, 513, 1026, 0, 16388, 57345, 16389, 196610, 262149, 0, 786438, 3145733, 6291461, 0, 29360132, 0, 67108871, 234881030, 201326601, 1207959557, 2415919109, 3758096387, 5368709130, 10737418245, 30064771083
Offset: 1

Views

Author

David W. Wilson

Keywords

Links

Crossrefs

Cf. A162216, A162217, A162218. In other bases: A033836 (base 4), A033837 (base 5), A033838 (base 6), A033839 (base 7), A033840 (base 8), A033841 (base 9), A003321 (base 10). - Joseph Myers, Jul 07 2009

A162218 Greatest integer equal to the sum of the n-th powers of its base-3 digits (written in base 10).

Original entry on oeis.org

1, 2, 8, 17, 1, 65, 131, 512, 259, 2050, 4100, 1, 16388, 57346, 65539, 196610, 458755, 1, 1572868, 3670019, 6291461, 1, 41943041, 1, 100663298, 234881030, 335544328, 1207959557, 2415919109, 4831838216, 5368709131, 10737418246, 34359738373
Offset: 0

Views

Author

Joseph Myers, Jun 28 2009

Keywords

Links

Crossrefs

Cf. A162216, A162217, A033835. In other bases: A162221 (base 4), A162224 (base 5), A162227 (base 6), A162230 (base 7), A162233 (base 8), A162236 (base 9), A046761 (base 10).

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

Views

Author

Joseph Myers, Jun 28 2009

Keywords

Comments

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

Links

Crossrefs

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).

Programs

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

Views

Author

Joseph Myers, Jun 28 2009

Keywords

Comments

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

Links

Crossrefs

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).

Programs

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

Views

Author

Joseph Myers, Jun 28 2009

Keywords

Comments

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

Links

Crossrefs

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).

Programs

  • PARI
    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

Views

Author

Joseph Myers, Jun 28 2009

Keywords

Comments

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

Links

Crossrefs

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).

Programs

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

Views

Author

Joseph Myers, Jun 28 2009

Keywords

Comments

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

Links

Crossrefs

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).

Programs

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

Views

Author

Joseph Myers, Jun 28 2009

Keywords

Comments

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

Links

Crossrefs

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).

Programs

  • PARI
    select( {is_A162234(n, b=9)=nn|| return(t==n))}, [0..10^5]) \\ M. F. Hasler, Nov 21 2019
Showing 1-10 of 10 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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