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.

User: Gaurav Kumar

Gaurav Kumar's wiki page.

Gaurav Kumar has authored 26 sequences. Here are the ten most recent ones:

A164837 a(n) is a prime number such that every k-digit substring ( k < n ) taken from the left, is divisible by k.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 127, 149, 163, 167, 181, 223, 227, 229, 241, 263, 269, 281, 283, 307, 347, 349, 367, 383, 389, 401, 409, 421, 443, 449, 461, 463, 467, 487, 503, 509
Offset: 1

Views

Author

Gaurav Kumar, Aug 28 2009

Keywords

Examples

			269 is a member since 2 is divisible by 1 and 26 is divisible by 2.

A164882 Numbers n such that sum of the cubes of the digits of n^3 is a perfect cube.

Original entry on oeis.org

0, 1, 2, 10, 20, 78, 100, 118, 200, 237, 303, 501, 780, 1000, 1005, 1039, 1079, 1145, 1180, 1562, 2000, 2224, 2370, 3003, 3030, 3304, 4105, 4339, 5001, 5010, 5235, 5478, 5705, 5829, 6096, 6459, 6475, 6493, 6795, 7154, 7221, 7444, 7800, 8494, 9168, 9462
Offset: 1

Views

Author

Gaurav Kumar, Aug 29 2009

Keywords

Examples

			a(6) = 78 since 78^3 = 474552 and 4^3 + 7^3 + 4^3 + 5^3 + 5^3 + 2^3 = 729 = 9^3

Links

Programs

  • Mathematica
    Select[Range[0,10000],IntegerQ[Power[Total[IntegerDigits[#^3]^3], (3)^-1]]&] (* Harvey P. Dale, Mar 17 2012 *)

Extensions

Typo in name fixed by Zak Seidov, May 21 2013

A163502 a(n) is the smallest number which is divisible by n, is not equal to n and its digital sum is also divisible by n.

Original entry on oeis.org

2, 4, 6, 8, 50, 24, 70, 80, 18, 190, 209, 48, 247, 266, 195, 448, 476, 198, 874, 3980, 399, 2398, 1679, 888, 4975, 1898, 999, 7588, 4988, 39990, 8959, 17888, 42999, 28798, 57995, 29988, 37999, 59888, 49998, 699880, 177899, 88998, 99889, 479996, 499995
Offset: 1

Views

Author

Gaurav Kumar, Jul 29 2009

Keywords

Examples

			a(5) = 50 because 50 is the smallest number(not equal to the number itself) which is divisible by 5 and its digital sum is divisible by 5.

Programs

  • Mathematica
    sndn[n_]:=Module[{k=2},While[!Divisible[Total[IntegerDigits[k*n]],n], k++]; k*n]; Array[sndn,50] (* Harvey P. Dale, Feb 27 2015 *)
  • PARI
    a(n) = {k = 2; while (sumdigits(k*n) % n, k++); k*n;} \\ Michel Marcus, Oct 09 2013

A163390 Numbers n such that n^3 can be represented as sum of (at least two) consecutive squares.

Original entry on oeis.org

47, 65, 921, 2161, 2820, 12284, 13156, 16761, 18340, 43500, 61721, 63765, 64605, 66317, 99359, 116180, 122009, 146821, 159371, 260165, 269588, 314919, 403130, 524095, 690381, 813340, 827209, 869241, 985864, 1203015, 1348761, 1942084, 3145712, 3632486, 3867745
Offset: 1

Views

Author

Gaurav Kumar, Jul 26 2009

Keywords

Examples

			a(1) = 47 as 47^3 = 103823 = 22^2 + 23^2 + 24^2 + ... + 68^2.

Links

Formula

a(n)= (A136312(n))^(1/3). [R. J. Mathar, Aug 02 2009]

Extensions

a(6)-a(14) from Donovan Johnson, Feb 02 2011
a(15)-a(16) from Donovan Johnson, Jun 12 2011
a(17) from Donovan Johnson, Aug 02 2013
a(18)-a(35) from Chai Wah Wu, Jan 16 2016

A163392 Squares which can be represented as sum of (at least two) consecutive cubes and are not triangular numbers squared.

Original entry on oeis.org

41616, 97344, 99225, 104329, 254016, 345744, 518400, 4322241, 4708900, 8643600, 19998784, 24147396, 26904969, 34574400, 35808256, 43956900, 56205009, 76055841, 87609600, 114383025, 121528576, 171714816
Offset: 1

Views

Author

Gaurav Kumar, Jul 26 2009

Keywords

Examples

			a(1) = 204^2 = 41616 = 23^3 + 24^3 + 25^3.

Links

Formula

a(n) = (A163393(n))^2. [R. J. Mathar, Aug 03 2009]

Extensions

77053284 and 106007616 removed by Chai Wah Wu, Mar 10 2016

A163393 Numbers n such that n^2 can be represented as sum of (at least two) consecutive cubes and n is not a triangular number.

Original entry on oeis.org

204, 312, 315, 323, 504, 588, 720, 2079, 2170, 2940, 4472, 4914, 5187, 5880, 5984, 6630, 7497, 8721, 9360, 10695, 11024, 13104, 14160, 16296, 16380, 18333, 18810, 22022, 22330, 23247, 31248, 36729, 42021, 43065, 43309, 49665
Offset: 1

Views

Author

Gaurav Kumar, Jul 26 2009

Keywords

Comments

A subsequence of A126200. [R. J. Mathar, Aug 02 2009]
A supersequence of A238099. - N. J. A. Sloane, Feb 25 2014

Examples

			a(1) = 204 as 204^2 = 41616 = 23^3 + 24^3 + 25^3.

References

  • H. E. Dudeney, Amusements in Mathematics, Nelson, London, 1917, Problem 135.

Links

Crossrefs

Cf. A126200, A238099.

Extensions

8778 and 10296 removed by Chai Wah Wu, Mar 10 2016

A163245 Squares which can be represented as sum of (at least two) consecutive primes in exactly one way.

Original entry on oeis.org

49, 121, 144, 169, 324, 484, 625, 900, 1089, 1156, 1369, 1444, 1681, 2116, 2916, 3249, 3600, 3844, 4225, 4900, 5184, 5329, 6241, 6889, 7396, 7744, 8100, 8649, 8836, 9025, 10201, 11449, 11881, 12100, 12769, 14161, 14400, 14884, 15129, 16384, 17689
Offset: 1

Views

Author

Gaurav Kumar, Jul 23 2009

Keywords

Examples

			49 = 13 + 17 + 19 But not 36 since 36 = 5 + 7 + 11 + 13 = 17 + 19.

Extensions

Offset corrected by Arkadiusz Wesolowski, Mar 28 2012

A164836 a(n) is the number such that every k-digit substring ( k <= n ) taken from the right end, is divisible by k.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, 102, 108, 114, 120, 126, 132, 138, 144, 150, 156, 162, 168, 174, 180
Offset: 1

Views

Author

Gaurav Kumar, Aug 28 2009

Keywords

Examples

			144 is a member of the sequence since 4 is divisible by 1, 44 is divisible by 2 and 144 is divisible by 3.
For 30-digit number 177723702519951630930135507600 we have: 1 | 0, 2 | 0, 3 | 600, 4 | 7600, 5 | 7600, 6 | 507600, ... [_Jaroslav Krizek_, Feb 19 2010]

Programs

  • Mathematica
    ekdsQ[n_]:=Module[{idn=IntegerDigits[n]},And@@Table[Divisible[ FromDigits[ Take[ idn,-k]],k],{k,IntegerLength[n]}]]; Select[Range[0,200],ekdsQ] (* Harvey P. Dale, Nov 11 2017 *)

A163497 Numbers n with following property: let c = nearest cube to n that is different from n and let p = nearest prime to n that is different from n. Then |n-c| = |n-p|.

Original entry on oeis.org

2, 25, 28, 119, 126, 340, 345, 728, 731, 1329, 1346, 2188, 2200, 3374, 3382, 4911, 4916, 6858, 6861, 9259, 9269, 12165, 12182, 15622, 15627, 19682, 19685, 24384, 24390, 29790, 29797, 35935, 35944, 42869, 42887, 50652, 50662, 59300, 59326
Offset: 1

Views

Author

Gaurav Kumar, Jul 29 2009

Keywords

Comments

With the exception of 2 those k where A051699(k) = A074989(k) (same distance to nearest prime and to nearest cube). - R. J. Mathar, Aug 08 2009

Examples

			a(1) = 2 since 2 lies between 1 (cube) and 3 (prime);
a(2) = 28 since 28 lies between 27 (cube) and 29 (prime).

Crossrefs

Cf. A154840.

Programs

  • Maple
    A163497 := proc(n) option remember ; local a; if n = 1 then 2; else for a from procname(n-1)+1 do if A051699(a) = A074989(a) then return a; end if; end do ; end if; end proc: # R. J. Mathar, Nov 01 2009

Extensions

Edited by Zak Seidov, Aug 01 2009
Further edited by N. J. A. Sloane, Oct 31 2009

A159469 Maximum remainder when (k + 1)^n + (k - 1)^n is divided by k^2 for variable n and k > 2.

Original entry on oeis.org

6, 8, 20, 24, 42, 48, 72, 80, 110, 120, 156, 168, 210, 224, 272, 288, 342, 360, 420, 440, 506, 528, 600, 624, 702, 728, 812, 840, 930, 960, 1056, 1088, 1190, 1224, 1332, 1368, 1482, 1520, 1640, 1680, 1806, 1848, 1980, 2024, 2162, 2208, 2352, 2400, 2550, 2600
Offset: 3

Views

Author

Gaurav Kumar, Apr 13 2009

Keywords

Examples

			For n = 3, maxr => 3*3 - 3 = 6 since 3 is odd.
For n = 4, maxr => 4*4 - 2*4 = 8 since 4 is even.

Links

Crossrefs

Cf. A050187.

Programs

  • Mathematica
    LinearRecurrence[{1,2,-2,-1,1},{6,8,20,24,42},50] (* Harvey P. Dale, Apr 18 2018 *)
  • PARI
    a(n) = if (n % 2, n^2 - n, n^2 - 2*n); \\ Michel Marcus, Aug 26 2013
  • PARI
    first(n) = Vec(x^3*(-6-2*x)/((x+1)^2*(x-1)^3) + O(x^(n+3))) \\ Iain Fox, Nov 26 2017

Formula

maxr(n) = n*n - 2*n if n is even, and n*n - n if n is odd.
G.f.: x^3*(-6-2*x)/((x+1)^2*(x-1)^3). - Maksym Voznyy (voznyy(AT)mail.ru), Jul 26 2009 (proved by Iain Fox, Nov 26 2017)
a(n) = 2*A050187(n). - R. J. Mathar, Aug 08 2009 (proved by Iain Fox, Nov 27 2017)
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5) for n > 7. - Colin Barker, Oct 29 2017 (proved by Iain Fox, Nov 26 2017)
a(n) = n^2 - n*(3 + (-1)^n)/2. - Iain Fox, Nov 26 2017
From Iain Fox, Nov 27 2017: (Start)
a(n) = A000290(n) - A022998(n).
a(n) = 2*A093005(n-2) + A168273(n-1).
a(n) = (4*(A152749(n-2)) + A091574(n-1) - A010719(n-1))/3.
E.g.f.: x*(exp(x)*x - sinh(x)).
(End)

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