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 12 results. Next

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

A003321 Smallest n-th order perfect digital invariant or PDI: smallest number > 1 equal to sum of n-th powers of its digits, or 0 if no such number exists.

Original entry on oeis.org

2, 0, 153, 1634, 4150, 548834, 1741725, 24678050, 146511208, 4679307774, 32164049650, 0, 564240140138, 28116440335967, 0, 4338281769391370, 233411150132317, 0, 1517841543307505039, 63105425988599693916
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

Except for the initial term, this is the third column of A252648. - M. F. Hasler, Feb 16 2015
a(n) = 0 if n>1 and in A262094. - Dmitry Kamenetsky, Jun 05 2020

Examples

			1^3 + 5^3 + 3^3 = 153.
1*0^17 + 5*1^17 + 2*2^17 + 4*3^17 + 1*4^17 + 1*5^17 + 1*7^17 = 233411150132317.

References

  • M. Gardner, The Magic Numbers of Dr Matrix. Prometheus, Buffalo, NY, 1985, p. 249.
  • J. S. Madachy, Mathematics on Vacation, Thomas Nelson and Sons Ltd. 1966, p. 164.
  • J. S. Madachy, Madachy's Mathematical Recreations, Dover, p. 164.
  • C. A. Pickover, Keys to Infinity. New York: W. H. Freeman, pp. 169-170, 1995.
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A001694, A007532, A005934, A005188, A014576, A023052, A046074, A046761.
In other bases: A033835 (base 3), A033836 (base 4), A033837 (base 5), A033838 (base 6), A033839 (base 7), A033840 (base 8), A033841 (base 9).

Programs

  • PARI
    a(n)=m=1;while(m*9^n>=10^m,m++);for(k=2,10^m,d=digits(k);s=sum(i=1,#d,d[i]^n);if(s==k,return(k)));0
n=1;while(n<10,print1(a(n),", ");n++) \\ Derek Orr, Dec 19 2014

Extensions

Additional comments from Lekraj Beedassy, May 23 2001
Extended and cross-references edited by Joseph Myers, Jun 28 2009

A046074 Power of digits of A023052 such that the sum of powers of digits equals A023052.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 3, 3, 3, 4, 5, 5, 4, 4, 5, 5, 5, 5, 6, 7, 7, 7, 7, 7, 8, 8, 8, 9, 9, 9, 9, 10, 11, 11, 11, 11, 11, 11, 11, 11, 13, 14, 17, 16, 16, 17, 17, 17, 19, 19, 19, 19, 20, 21, 21, 23, 23, 23, 23, 23, 25, 24, 24, 24, 25, 25, 25, 25, 25, 25, 25, 27, 27, 27, 27, 27, 27
Offset: 1

Views

Author

Eric W. Weisstein

Keywords

Links

Crossrefs

Cf. A023052, A003321, A046761.
In other bases: A162217 (base 3), A162220 (base 4), A162223 (base 5), A162226 (base 6), A162229 (base 7), A162232 (base 8), A162235 (base 9).

Extensions

Offset corrected by N. J. A. Sloane, Jun 20 2009
More terms calculated by Joseph S. Myers. Hans Havermann, Jun 18 2009
Cross-references edited by Joseph Myers, Jun 28 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).

A162221 Greatest integer equal to the sum of the n-th powers of its base-4 digits (written in base 10).

Original entry on oeis.org

1, 3, 1, 62, 243, 33, 922, 2699, 13125, 40393, 1, 2049, 1075174, 6401871, 1, 57592238, 129336777, 516822801, 1163047905, 524289, 1, 62770507831, 1, 8388609, 1, 847624153770, 1, 53379719265825, 183016755558795, 411784948543867
Offset: 0

Views

Author

Joseph Myers, Jun 28 2009

Keywords

Links

Crossrefs

Cf. A162219, A162220, A033836. In other bases: A162218 (base 3), A162224 (base 5), A162227 (base 6), A162230 (base 7), A162233 (base 8), A162236 (base 9), A046761 (base 10).

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

Views

Author

Joseph Myers, Jun 28 2009

Keywords

Links

Crossrefs

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).

A162227 Greatest integer equal to the sum of the n-th powers of its base-6 digits (written in base 10).

Original entry on oeis.org

1, 5, 1, 251, 1, 6197, 36140, 269458, 391907, 10067135, 1, 1, 1, 6170640875, 1, 124246559501, 1, 5341093125745, 19418246235419, 58598336876363, 196242885967642, 3359860808719207, 1, 36185564197761935
Offset: 0

Views

Author

Joseph Myers, Jun 28 2009

Keywords

Links

Crossrefs

Cf. A162225, A162226, A033838. In other bases: A162218 (base 3), A162221 (base 4), A162224 (base 5), A162230 (base 7), A162233 (base 8), A162236 (base 9), A046761 (base 10).

A162230 Greatest integer equal to the sum of the n-th powers of its base-7 digits (written in base 10).

Original entry on oeis.org

1, 6, 45, 250, 1, 9286, 35411, 794376, 72865, 34403018, 9885773, 1500022495, 1, 28700208851, 1, 2352056093102, 1, 102340463411217, 1, 3123368686057682, 1, 132116164569671440, 402070508232683666, 5528252581301500133
Offset: 0

Views

Author

Joseph Myers, Jun 28 2009

Keywords

Links

Crossrefs

Cf. A162228, A162229, A033839. In other bases: A162218 (base 3), A162221 (base 4), A162224 (base 5), A162227 (base 6), A162233 (base 8), A162236 (base 9), A046761 (base 10).

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

Views

Author

Joseph Myers, Jun 28 2009

Keywords

Links

Crossrefs

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).

A162236 Greatest integer equal to the sum of the n-th powers of its base-9 digits (written in base 10).

Original entry on oeis.org

1, 8, 50, 1052, 1824, 65539, 36804, 5300099, 6287267, 156608074, 3302332571, 177148, 155896317510, 1212041747339, 2113843488367, 5752216689703, 320659684133768, 6774649666149786, 37860400025315399, 157971147790033100
Offset: 0

Views

Author

Joseph Myers, Jun 28 2009

Keywords

Links

Crossrefs

Cf. A162234, A162235, A033841. In other bases: A162218 (base 3), A162221 (base 4), A162224 (base 5), A162227 (base 6), A162230 (base 7), A162233 (base 8), A046761 (base 10).
Showing 1-10 of 12 results. Next

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

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