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 1-5 of 5 results.

A071849 Numbers k such that A245788(k) > prime(k).

Original entry on oeis.org

3, 7, 11, 15, 23, 27, 29, 30, 31, 47, 55, 59, 61, 62, 63, 95, 111, 119, 123, 125, 126, 127, 159, 175, 183, 187, 189, 191, 223, 239, 247, 251, 253, 254, 255, 319, 351, 367, 375, 379, 381, 382, 383, 415, 431, 439, 447, 479, 495, 503, 507, 509, 510, 511, 639, 703
Offset: 1

Views

Author

Benoit Cloitre, Jun 09 2002

Keywords

Links

Crossrefs

Cf. A000040, A000120, A245788.

Programs

  • Maple
    p:= 1: Res:= NULL: count:= 0:
for n from 1 while count < 100 do
  p:= nextprime(p);
  if n*convert(convert(n,base,2),`+`) > p then
    count:= count+1; Res:= Res, n;
  fi
od:
Res; # Robert Israel, Oct 30 2018
  • Mathematica
    Select[Range[700], # * DigitCount[#, 2, 1] > Prime[#] &] (* Amiram Eldar, Jun 03 2022 *)
  • PARI
    for(n=1,1000,b=binary(n); if(sum(i=1,length(b), component(b,i))*n>prime(n),print1(n,",")))
  • PARI
    is(n,p=prime(n))=hammingweight(n)*n>p \\ Charles R Greathouse IV, Oct 30 2018
  • PARI
    list(lim)=my(v=List(),n); forprime(p=2,, if(n++>lim, return(Vec(v))); if(hammingweight(n)*n>p, listput(v,n))); \\ Charles R Greathouse IV, Oct 30 2018

A057147 a(n) = n times sum of digits of n.

Original entry on oeis.org

0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 10, 22, 36, 52, 70, 90, 112, 136, 162, 190, 40, 63, 88, 115, 144, 175, 208, 243, 280, 319, 90, 124, 160, 198, 238, 280, 324, 370, 418, 468, 160, 205, 252, 301, 352, 405, 460, 517, 576, 637, 250, 306, 364, 424, 486, 550, 616
Offset: 0

Views

Author

N. J. A. Sloane, Sep 13 2000

Keywords

Comments

A056992(n) = A010888(a(n)). - Reinhard Zumkeller, Mar 19 2014

Links

Crossrefs

Cf. A007953, A003634, A005349, A052489, A052490, A003635, A245788.
Iterations: A047892 (start=2), A047912 (start=3), A047897 (start=5), A047898 (start=6), A047899 (start=7), A047900 (start=8), A047901 (start=9), A047902 (start=11).

Programs

  • Haskell
    a057147 n = a007953 n * n  -- Reinhard Zumkeller, Mar 19 2014
  • Maple
    for n from 0 to 150 do printf(`%d,`,n*add(convert(n, base, 10)[i], i=1..nops(convert(n,base, 10)))) od:
  • Mathematica
    Table[n*Total[IntegerDigits[n]], {n, 0, 100}]
  • PARI
    a(n) = n*sumdigits(n) \\ Franklin T. Adams-Watters, Aug 03 2014
  • Python
    [n*sum([int(d) for d in str(n)]) for n in range(10**5)] # Chai Wah Wu, Aug 05 2014

Formula

a(n) = n*A007953(n). - Michel Marcus, Aug 10 2014
G.f.: x * (d/dx) (1/(1 - x))*Sum_{k>=1} (x^k - x^(10^k+k) - 9*x^(10^k))/(1 - x^(10^k)). - Ilya Gutkovskiy, Mar 27 2018

Extensions

More terms from James Sellers and Larry Reeves (larryr(AT)acm.org), Sep 13 2000

A249154 (n+1) times the number of 1's in the binary expansion of n.

Original entry on oeis.org

0, 2, 3, 8, 5, 12, 14, 24, 9, 20, 22, 36, 26, 42, 45, 64, 17, 36, 38, 60, 42, 66, 69, 96, 50, 78, 81, 112, 87, 120, 124, 160, 33, 68, 70, 108, 74, 114, 117, 160, 82, 126, 129, 176, 135, 184, 188, 240, 98, 150, 153, 208, 159, 216, 220, 280, 171, 232, 236, 300, 244, 310, 315, 384, 65
Offset: 0

Views

Author

Antti Karttunen, Nov 02 2014

Keywords

Links

Crossrefs

Cf. A000120, A000788, A187059, A245788, A091512.

Programs

Formula

a(n) = (n+1) * A000120(n).
a(n) = A000120(n) + A245788(n).
a(n) = 2*A000788(n) - A187059(n).

A302451 a(n) = [x^n] Product_{k>=0} (1 + n*x^(2^k)).

Original entry on oeis.org

1, 1, 2, 9, 4, 25, 36, 343, 8, 81, 100, 1331, 144, 2197, 2744, 50625, 16, 289, 324, 6859, 400, 9261, 10648, 279841, 576, 15625, 17576, 531441, 21952, 707281, 810000, 28629151, 32, 1089, 1156, 42875, 1296, 50653, 54872, 2313441, 1600, 68921, 74088, 3418801, 85184, 4100625, 4477456, 229345007, 2304
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 08 2018

Keywords

Examples

			+---+-----+---+----------+
| n | bin.|1's|   a(n)   |
+---+-----+---+----------+
| 0 |   0 | 0 | 0^0 =  1 |
| 1 |   1 | 1 | 1^1 =  1 |
| 2 |  10 | 1 | 2^1 =  2 |
| 3 |  11 | 2 | 3^2 =  9 |
| 4 | 100 | 1 | 4^1 =  4 |
| 5 | 101 | 2 | 5^2 = 25 |
| 6 | 110 | 2 | 6^2 = 36 |
+---+-----+---+----------+
bin. - n written in base 2;
1's - number of 1's in binary expansion of n.

Links

Crossrefs

Main diagonal of A256140.
Cf. A000120, A001316, A048883, A102376, A245788.

Programs

  • Mathematica
    Table[SeriesCoefficient[Product[(1 + n x^(2^k)), {k, 0, n}], {x, 0, n}], {n, 0, 48}]
Join[{1}, Table[n^DigitCount[n, 2, 1], {n, 48}]]
  • PARI
    a(n) = n^hammingweight(n); \\ Altug Alkan, Apr 08 2018

Formula

a(n) = n^A000120(n).
a(n) = A256140(n,n).
a(2^k) = 2^k.
a(2^k-1) = (2^k - 1)^k.

A301895 a(n) = (number of 1's in binary expansion of n)^(number of 0's in binary expansion of n).

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 2, 1, 1, 4, 4, 3, 4, 3, 3, 1, 1, 8, 8, 9, 8, 9, 9, 4, 8, 9, 9, 4, 9, 4, 4, 1, 1, 16, 16, 27, 16, 27, 27, 16, 16, 27, 27, 16, 27, 16, 16, 5, 16, 27, 27, 16, 27, 16, 16, 5, 27, 16, 16, 5, 16, 5, 5, 1, 1, 32, 32, 81, 32, 81, 81, 64, 32, 81, 81, 64, 81, 64, 64, 25, 32
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 28 2018

Keywords

Comments

Union of A000079 and A000225 without zero gives positions of ones.

Examples

			+---+------+---+---+---------+
| n | bin. |1's|0's|  a(n)   |
+---+------+---+---+---------+
| 0 |    0 | 0 | 1 | 0^1 = 0 |
| 1 |    1 | 1 | 0 | 1^0 = 1 |
| 2 |   10 | 1 | 1 | 1^1 = 1 |
| 3 |   11 | 2 | 0 | 2^0 = 1 |
| 4 |  100 | 1 | 2 | 1^2 = 1 |
| 5 |  101 | 2 | 1 | 2^1 = 2 |
| 6 |  110 | 2 | 1 | 2^1 = 2 |
| 7 |  111 | 3 | 0 | 3^0 = 1 |
| 8 | 1000 | 1 | 3 | 1^3 = 1 |
| 9 | 1001 | 2 | 2 | 2^2 = 4 |
+---+------+---+---+---------+
bin. - n written in base 2;
1's - number of 1's in binary expansion of n;
0's - number of 0's in binary expansion of n.

Links

Crossrefs

Cf. A000051, A000079, A000120, A000225, A011782, A023416, A037861, A070939, A071295, A245788.

Programs

  • Mathematica
    DigitCount[Range[0, 80], 2, 1]^DigitCount[Range[0, 80], 2, 0]

Formula

a(n) = A000120(n)^A023416(n).
a(A000051(n)) = A011782(n).
Showing 1-5 of 5 results.

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

Content is available under The OEIS End-User License Agreement.