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 11-17 of 17 results.

A055591 a(n) = a(n-1)+3^a(n-1).

Original entry on oeis.org

0, 1, 4, 85, 35917545547686059365808220080151141317128
Offset: 0

Views

Author

Henry Bottomley, May 26 2000

Keywords

Examples

			a(3) = 4+3^4 = 4+81 = 85

Crossrefs

Cf. A034797, A055590.

Programs

Formula

a(n) = A055591(n-1)+A055590(n)

A105035 Positions of record values in A104234.

Original entry on oeis.org

0, 1, 5, 2037
Offset: 1

Views

Author

N. J. A. Sloane, Apr 04 2005

Keywords

Comments

Is this 2^g(n-1) - g(n-1), where g = A034797?
Yes, this formula can be proved by induction on n. - Max Alekseyev, Mar 16 2023
a(5) = 2^2059 - 2059 with 620 decimal digits is too large to be included here.

Links

  • David Applegate, Benoit Cloitre, Philippe Deléham and N. J. A. Sloane, Sloping binary numbers: a new sequence related to the binary numbers [pdf, ps].

Formula

a(n) = 2^A034797(n-1) - A034797(n-1). - Max Alekseyev, Mar 16 2023

Extensions

Offset corrected by Max Alekseyev, Mar 17 2023

A309274 Ackermann Coding (BIT predicate) of transitive hereditarily finite sets.

Original entry on oeis.org

0, 1, 3, 7, 11, 15, 23, 31, 39, 47, 55, 63, 71, 79, 87, 95, 103, 111, 119, 127, 135, 143, 151, 159, 167, 175, 183, 191, 199, 207, 215, 223, 231, 239, 247, 255, 267, 271, 287, 303, 319, 335, 351, 367, 383, 399, 415, 431, 447, 463, 479, 495, 511, 523, 527, 543
Offset: 1

Views

Author

Christophe Papazian, Jul 24 2019

Keywords

Comments

If the representation of a(n) in base 2 contains the k-th bit (2^k), then it must contain the bits of k.
A034797 is a subsequence, and can be seen as a recursive variant of this sequence. - Rémy Sigrist, Jul 25 2019

Examples

			23 is in the sequence because 23 = 2^4 + 2^2 + 2^1 + 2^0 encodes the transitive set {0,1,{1},{{1}}} (remember that 0 is the empty set and 1 is {0}).

Links

Crossrefs

Cf. A034797, A102553.

Programs

  • Mathematica
    b[n_] := (Flatten @ Position[Reverse[IntegerDigits[n, 2]], 1] - 1);
okQ[n_] := With[{bb = b[n]}, AllTrue[b /@ bb, Intersection[bb, #] == #&]];
Select[Range[0, 600], okQ] (* Jean-François Alcover, Jul 25 2019 *)
  • PARI
    is(n) = { for (b=0, #binary(n), if (bittest(n, b), if (bitand(n, b)!=b, return (0)))); return (1) } \\ Rémy Sigrist, Jul 25 2019

A054874 a(n) = 2^(sum of a(i) where i

Original entry on oeis.org

0, 1, 2, 8, 2048
Offset: 0

Views

Author

Henry Bottomley, May 26 2000

Keywords

Comments

The next term is too large to include.

Examples

			a(4) = 2^(0+1+2+8) = 2^11 = 2048; a(5) = 2^2059>10^619

Crossrefs

Cf. A014221, A034797 for partial sum, so a(n) is number of impartial games with value n-1, using natural enumeration of impartial games.

Formula

a(n) = 2^A034797(n-1) = A034797(n) - A034797(n-1)

A327777 Prime numbers whose binary indices have integer mean and integer geometric mean.

Original entry on oeis.org

2, 257, 8519971, 36574494881, 140739702949921, 140773995710729, 140774004099109
Offset: 1

Views

Author

Gus Wiseman, Sep 27 2019

Keywords

Comments

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.
Conjecture: This sequence is infinite.

Examples

			The initial terms together with their binary indices:
                2: {2}
              257: {1,9}
          8519971: {1,2,6,9,18,24}
      36574494881: {1,6,8,16,18,27,32,36}
  140739702949921: {1,6,12,27,32,48}
  140773995710729: {1,4,9,12,18,32,36,48}
  140774004099109: {1,3,6,12,18,24,32,36,48}

Links

Crossrefs

A subset of A327368.
The binary weight of prime(n) is A014499(n), with binary length A035100(n).
Heinz numbers of partitions with integer mean: A316413.
Heinz numbers of partitions with integer geometric mean: A326623.
Heinz numbers with both: A326645.
Subsets with integer mean: A051293
Subsets with integer geometric mean: A326027
Subsets with both: A326643
Partitions with integer mean: A067538
Partitions with integer geometric mean: A067539
Partitions with both: A326641
Strict partitions with integer mean: A102627
Strict partitions with integer geometric mean: A326625
Strict partitions with both: A326029
Factorizations with integer mean: A326622
Factorizations with integer geometric mean: A326028
Factorizations with both: A326647
Numbers whose binary indices have integer mean: A326669
Numbers whose binary indices have integer geometric mean: A326673
Numbers whose binary indices have both: A327368
Cf. A000120, A029931, A034797, A048793, A070939, A096111, A291166, A326031, A326644, A326699/A326700.

Programs

  • Mathematica
    bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Select[Prime[Range[1000]],IntegerQ[Mean[bpe[#]]]&&IntegerQ[GeometricMean[bpe[#]]]&]

Extensions

a(4)-a(7) from Giovanni Resta, Dec 01 2019

A105553 a(n) is the number of 1's in A103528(n) written in base 2.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 1, 0, 1, 1, 2, 0, 1, 1, 1, 0, 1, 1, 2, 1, 1, 1, 1, 0, 1, 1, 2, 0, 1, 1, 1, 0, 1, 1, 2, 1, 2, 1, 1, 0, 1, 1, 2, 0, 1, 1, 1, 0, 1, 1, 2, 1, 1, 1, 1, 0, 1, 1, 2, 0, 1, 1, 1, 0, 1, 1, 2, 1, 2, 2, 1, 0, 1, 1, 2, 0, 1, 1, 1, 0, 1, 1, 2, 1, 1, 1, 1, 0, 1, 1, 2, 0, 1, 1, 1, 0, 1, 1, 2
Offset: 1

Views

Author

Philippe Deléham, May 03 2005

Keywords

Links

  • David Applegate, Benoit Cloitre, Philippe Deléham and N. J. A. Sloane, Sloping binary numbers: a new sequence related to the binary numbers [pdf, ps].

Crossrefs

For records see A034797. Cf. A103318, A103528.

Formula

a(n) = A103318(n) - 1.

A333132 a(n) = n for n <= 3; thereafter a(n) = 2^(a(n-1)-1) + a(n-1).

Original entry on oeis.org

1, 2, 3, 7, 71, 1180591620717411303495
Offset: 1

Views

Author

Ilya Gutkovskiy, Mar 08 2020

Keywords

Comments

The next term is too large to include.
a(n) = number of compositions of a(1) + number of compositions of a(2) + ... + number of compositions of a(n-1) for n > 2.

Examples

			a(5) = 71, 71 in base 2 (reverse order of digits) = 1110001.
                                                    |||   |
                                                    123   7

Crossrefs

Cf. A004249, A007516, A011782, A014221, A034797, A103527.

Programs

  • Mathematica
    a[n_] := a[n] = If[n <= 3, n, 2^(a[n - 1] - 1) + a[n - 1]]; Table[a[n], {n, 1, 6}]

Formula

a(n) = n for n <= 2; thereafter a(n) = Sum_{k=1..n-1} 2^(a(k)-1).
Previous Showing 11-17 of 17 results.

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

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