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.

Previous Showing 11-19 of 19 results.

A135104 Digital sum (base the n-th prime) of n^4.

Original entry on oeis.org

1, 4, 5, 10, 15, 24, 17, 28, 27, 60, 31, 36, 81, 70, 71, 68, 117, 96, 37, 120, 81, 100, 139, 192, 97, 176, 123, 174, 205, 128, 193, 126, 137, 220, 201, 216, 133, 196, 397, 296, 189, 396, 321, 256, 305, 280, 331, 396, 445, 292, 313, 256, 481, 556, 417, 326, 553, 256
Offset: 1

Views

Author

Hieronymus Fischer, Dec 24 2007

Keywords

Examples

			a(2) = ds_prime(2)(2^4) = ds_3(16) = 1+2+1 = 4;
a(6) = ds_prime(6)(6^4) = ds_13(1296) = 7+8+9 = 24.

Links

Crossrefs

Cf. A000040, A007953, A054899, A075771, A131451, A133620, A133900, A134599.

Programs

  • Mathematica
    Table[Total[IntegerDigits[n^4, Prime[n]]], {n, 50}] (* G. C. Greubel, Sep 23 2016 *)
  • PARI
    a(n) = vecsum(digits(n^4, prime(n))); \\ Michel Marcus, Sep 24 2016

Formula

a(n) = ds_prime(n)(n^4), where ds_prime(n) = digital sum base the n-th prime.
a(n) = n^4 - (prime(n)-1)*Sum{k>0} ( floor(n^4/prime(n)^k) ).

A135105 Digital sum (base the n-th prime) of n^5.

Original entry on oeis.org

1, 4, 11, 16, 15, 12, 23, 44, 45, 40, 41, 72, 93, 98, 99, 48, 133, 108, 109, 160, 117, 172, 81, 120, 217, 176, 159, 102, 221, 144, 187, 262, 169, 304, 375, 276, 241, 158, 211, 316, 273, 72, 313, 320, 397, 406, 227, 582, 335, 236, 187, 460, 293, 274, 663, 178, 433, 538
Offset: 1

Views

Author

Hieronymus Fischer, Dec 24 2007

Keywords

Examples

			a(2) = ds_prime(2)(2^5) = ds_3(32) = 1+0+1+2 = 4;
a(6) = ds_prime(5)(5^5) = ds_11(3125) = 2+9+3+1 = 15.

Links

Crossrefs

Cf. A000040, A007953, A054899, A075771, A131451, A133620, A133900, A134599.

Programs

  • Mathematica
    Table[Total[IntegerDigits[n^5,Prime[n]]],{n,60}] (* Harvey P. Dale, Jul 19 2013  *)
  • PARI
    a(n) = vecsum(digits(n^5, prime(n))); \\ Michel Marcus, Sep 24 2016

Formula

a(n) = ds_prime(n)(n^5), where ds_prime(n) = digital sum base the n-th prime.
a(n) = n^5 - (prime(n)-1)*Sum_{k>0} ( floor(n^5/prime(n)^k) ).

A135111 Numbers such that the digital sum base 2 and the digital sum base 3 are in a ratio of 2:3.

Original entry on oeis.org

5, 33, 78, 106, 116, 142, 150, 154, 156, 170, 178, 184, 202, 204, 210, 215, 226, 228, 278, 284, 294, 308, 312, 332, 338, 340, 344, 356, 377, 390, 396, 418, 420, 424, 450, 455, 467, 473, 483, 513, 526, 534, 550, 582, 586, 588, 596, 600, 612, 619, 624, 629
Offset: 1

Views

Author

Hieronymus Fischer, Dec 24 2007

Keywords

Examples

			a(1)=5, since ds_2(5):ds_3(5)=2:3.

Links

Crossrefs

Cf. A007953, A054899, A131451, A133620, A133900, A134599.

Programs

  • Maple
    select(t -> convert(convert(t,base,3),`+`)/convert(convert(t,base,2),`+`) = 3/2, [$1..1000]); # Robert Israel, Sep 26 2016
  • Mathematica
    Select[Range[500], 3*Total[IntegerDigits[#, 2]] == 2*Total[IntegerDigits[#, 3]] &] (* G. C. Greubel, Sep 26 2016 *)
  • PARI
    isok(n) = 2*vecsum(digits(n, 3)) == 3*vecsum(digits(n, 2)); \\ Michel Marcus, Sep 26 2016

A135123 Numbers such that the digital sum base 2 and the digital sum base 3 and the digital sum base 6 all are equal.

Original entry on oeis.org

1, 12, 13, 114, 115, 366, 367, 477, 687, 864, 865, 876, 877, 1086, 1087, 1305, 1326, 1327, 1386, 1387, 1596, 1597, 1626, 1627, 1656, 1657, 1746, 1747, 1836, 1837, 1956, 1957, 2595, 2607, 2646, 2647, 3276, 3277, 3906, 3907, 3948, 3949, 4068, 4069, 5438
Offset: 1

Views

Author

Hieronymus Fischer, Dec 31 2007

Keywords

Examples

			a(2)=12, since ds_2(12)=ds_3(12)=ds_6(12), where ds_x=digital sum base x.

Links

Crossrefs

Cf. A007953, A054899, A131451, A133620, A133900, A134599, A135100, A135110, A135120, A037308.

Programs

  • Mathematica
    Select[Range[5000], Total[IntegerDigits[#, 2]] == Total[IntegerDigits[#, 3]] == Total[IntegerDigits[#, 6]] &] (* G. C. Greubel, Sep 26 2016 *)

A135124 Numbers such that the digital sums in base 2, base 4 and base 8 are all equal.

Original entry on oeis.org

1, 64, 65, 4096, 4097, 4160, 4161, 262144, 262145, 262208, 262209, 266240, 266241, 266304, 266305, 16777216, 16777217, 16777280, 16777281, 16781312, 16781313, 16781376, 16781377, 17039360, 17039361, 17039424, 17039425, 17043456
Offset: 1

Views

Author

Hieronymus Fischer, Dec 31 2007, Dec 31 2008

Keywords

Comments

Written as base 64 numbers the sequence is 1,10,11,100,101,110,111,1000,1001, ... (cf. A007088)

Examples

			a(7)=4161, since ds_2(4161 )=ds_4(4161 )=ds_8(4161 ), where ds_x=digital sum base x.

Links

Crossrefs

Cf. A007953, A054899, A131451, A133620, A133900, A134599, A135100, A135110, A135120, A037308.
Cf. A000695, A033045.

Programs

  • Mathematica
    Select[Range[500000], Total[IntegerDigits[#, 2]] == Total[IntegerDigits[#, 4]] == Total[IntegerDigits[#, 8]] &] (* G. C. Greubel, Sep 26 2016 *)
With[{k = 64}, Rest@ Map[FromDigits[#, k] &, Tuples[{0, 1}, 5]]] (* Michael De Vlieger, Oct 28 2022 *)
Select[Range[171*10^5],Length[Union[Total/@IntegerDigits[#,{2,4,8}]]]==1&] (* Harvey P. Dale, May 14 2025 *)
  • PARI
    a(n) = fromdigits(binary(n),64); \\ Kevin Ryde, Apr 02 2025

Formula

a(n) = (1/2)*Sum_{k=0..floor(log_2(n))} (1-(-1)^floor(n/2^k))*64^k.
G.f.: (1/(1-x))*Sum_{k>=0} 64^k*x^(2^k)/(1+x^(2^k)).
a(n) = A000695(A033045(n)) = A033045(A000695(n)). - Alan Michael Gómez Calderón, Mar 27 2025

Extensions

Edited by N. J. A. Sloane, Jan 17 2009

A135125 Numbers such that the digital sum base 2 and the digital sum base 5 and the digital sum base 10 all are equal.

Original entry on oeis.org

1, 1300, 1301, 5010, 5011, 7102, 7103, 10050, 10051, 10235, 11135, 12250, 12251, 14015, 16102, 16103, 20060, 20061, 20206, 20207, 23230, 23231, 32012, 32013, 32302, 32303, 32410, 32411, 44000, 44001, 45010, 45011, 50012, 50013, 50300
Offset: 1

Views

Author

Hieronymus Fischer, Dec 31 2007

Keywords

Examples

			a(2)=1300, since ds_2(1300)=ds_5(1300)=ds_10(1300), where ds_x=digital sum base x.

Links

Crossrefs

Cf. A007953, A054899, A131451, A133620, A133900, A134599, A135100, A135110, A135120, A037308.

Programs

  • Mathematica
    Select[Range[10000], Total[IntegerDigits[#, 2]] == Total[IntegerDigits[#, 5]] == Total[IntegerDigits[#, 10]] &] (* G. C. Greubel, Sep 27 2016 *)

A135126 Numbers such that the digital sums in bases 3, 4, 5 and 6 all are equal.

Original entry on oeis.org

1, 2, 188, 668, 908, 1388, 1628, 2170, 2171, 2830, 2831, 3908, 4330, 4331, 6490, 6491, 8650, 8651, 10390, 10391, 10629, 12792, 12793, 12794, 17110, 17111, 17290, 17291, 25930, 25931, 36312, 36313, 36314, 37812, 37813, 37814, 41532, 41533, 41534
Offset: 1

Views

Author

Hieronymus Fischer, Dec 31 2007

Keywords

Examples

			a(3)=188, since ds_3(188)=ds_4(188)=ds_5(188)=ds_6(188)=8, where ds_x=digital sum base x.

Links

Crossrefs

Cf. A007953, A054899, A131451, A133620, A133900, A134599, A135100, A135110, A135120, A037308.

Programs

  • Mathematica
    Select[Range[3000], Total[IntegerDigits[#, 3]] == Total[IntegerDigits[#, 4]] ==  Total[IntegerDigits[#, 5]] == Total[IntegerDigits[#, 6]] &] (* G. C. Greubel, Sep 27 2016 *)

A135128 Numbers such that the digital sums in bases 2, 3, 5 and 10 all are equal.

Original entry on oeis.org

1, 12250, 12251, 23230, 23231, 32410, 32411, 45010, 45011, 51130, 51131, 52030, 52031, 54010, 54011, 100053, 100090, 100091, 100305, 102250, 102251, 107002, 107003, 110134, 110170, 110171, 110350, 110351, 110460, 110461, 113050, 113051
Offset: 1

Views

Author

Hieronymus Fischer, Dec 31 2007

Keywords

Examples

			a(2)=12250 since ds_2(12250 )=ds_3(12250 )=ds_5(12250 )=ds_10(12250 )=10, where ds_x=digital sum base x.

Links

Crossrefs

Cf. A007953, A054899, A131451, A133620, A133900, A134599, A135100, A135110, A135120, A037308.

Programs

  • Mathematica
    Select[Range[32000], Total[IntegerDigits[#, 2]] == Total[IntegerDigits[#, 3]] == Total[IntegerDigits[#, 5]] == Total[IntegerDigits[#, 10]] &] (* G. C. Greubel, Sep 28 2016 *)
Select[Range[120000],Length[Union[Total/@IntegerDigits[#,{2,3,5,10}]]]==1&] (* Harvey P. Dale, Mar 30 2024 *)

A135129 Numbers such that the digital sums in bases 3, 4, 5, 6 and 7 all are equal.

Original entry on oeis.org

1, 2, 1388, 2170, 2171, 2830, 2831, 10390, 10391, 12792, 12793, 12794, 17110, 17111, 17290, 17291, 36312, 36313, 36314, 37814, 41532, 41533, 41534, 50892, 50893, 50894, 52216, 52217, 52395, 56652, 56653, 56654, 95354, 96432, 96433, 96434
Offset: 1

Views

Author

Hieronymus Fischer, Dec 31 2007

Keywords

Examples

			a(3)=1388, since ds_3(1388)=ds_4(1388)=ds_5(1388)=ds_6(1388)=ds_7(1388), where ds_x=digital sum base x.

Links

Crossrefs

Cf. A007953, A054899, A131451, A133620, A133900, A134599, A135100, A135110, A135120, A037308.

Programs

  • Mathematica
    Select[Range[30000], Total[IntegerDigits[#, 3]] == Total[IntegerDigits[#, 4]] == Total[IntegerDigits[#, 5]] == Total[IntegerDigits[#, 6]] == Total[IntegerDigits[#, 7]] &] (* G. C. Greubel, Sep 28 2016 *)
Select[Range[100000],Length[Union[Table[Total[IntegerDigits[#,b]],{b,3,7}]]]==1&] (* Harvey P. Dale, Oct 25 2020 *)
Previous Showing 11-19 of 19 results.

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

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