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 1-4 of 4 results.

A292593 a(n) = A292591(A245612(n)).

Original entry on oeis.org

0, 1, 2, 2, 4, 5, 4, 5, 8, 8, 10, 11, 8, 8, 10, 10, 16, 17, 16, 17, 20, 20, 22, 23, 16, 17, 16, 16, 20, 21, 20, 21, 32, 32, 34, 35, 32, 32, 34, 34, 40, 41, 40, 41, 44, 44, 46, 47, 32, 32, 34, 34, 32, 33, 32, 32, 40, 40, 42, 43, 40, 40, 42, 42, 64, 65, 64, 65, 68, 68, 70, 71, 64, 65, 64, 64, 68, 69, 68, 69, 80, 80, 82, 83, 80, 80, 82, 82, 88, 89, 88, 89
Offset: 0

Views

Author

Antti Karttunen, Sep 20 2017

Keywords

Links

Crossrefs

Cf. A245612, A292591, A292592.
Differs from A292271 for the first time at n=31, where a(31) = 21, while A292271(31) = 20.

Programs

  • Mathematica
    f[n_] := f[n] = Which[n == 1, 0, Mod[n, 3] == 2, Ceiling[n/3], True, (Times @@ Power[If[# == 1, 1, NextPrime[#, -1]] & /@ First@ #, Last@ #] &@ Transpose@ FactorInteger[2 n - 1] + 1)/2]; g[n_] := g[n] = If[n <= 2, n - 1, 2 g[f@ n] + Boole[Mod[n, 3] == 1]]; h[n_] := (Times @@ Power[If[# == 1, 1, NextPrime@ #] & /@ First@ #, Last@ #] + 1)/2 &@ Transpose@ FactorInteger@ If[n == 0, 1, Prime[#] Product[Prime[m]^(Map[ Ceiling[ (Length@ # - 1)/2] &, DeleteCases[Split@ Join[Riffle[IntegerDigits[n, 2], 0], {0}], {k__} /; k == 1]][[-m]]), {m, #}] &[DigitCount[n, 2, 1]]]; Array[g@ h@ # &, 92, 0] (* Michael De Vlieger, Sep 22 2017 *)

Formula

a(n) + A292592(n) = n.

A292595 a(n) = A000120(A292591(n)).

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 3, 1, 3, 4, 2, 4, 2, 1, 4, 5, 2, 1, 6, 3, 6, 7, 1, 7, 3, 3, 7, 3, 4, 7, 8, 2, 2, 9, 4, 9, 10, 2, 1, 11, 1, 11, 4, 4, 11, 4, 5, 3, 12, 2, 12, 13, 1, 13, 14, 6, 14, 5, 3, 2, 4, 6, 1, 15, 7, 15, 5, 1, 15, 16, 7, 1, 5, 3, 16, 17, 3, 4, 18, 7, 3, 19, 3, 19, 5, 4, 19, 2, 7, 19, 20, 8, 5, 5, 2, 20, 21, 2, 21, 22, 9, 5, 7, 4, 2
Offset: 1

Views

Author

Antti Karttunen, Sep 20 2017

Keywords

Comments

If n > 1, then locate the node which contains n in binary tree A245612 (or in its mirror-image A244154) and traverse from that node towards the root [by iterating the map n -> A285712(n)], at the same time counting all numbers of the form 3k+1 that occur on the path, down to the final 1. This count includes also n itself if it is of the form 3k+1, with k > 0 (thus a(1) = 0).

Links

Crossrefs

Cf. A000120, A285712, A285715, A292591, A292594.

Programs

  • Scheme
    (define (A292595 n) (if (<= n 2) (- n 1) (+ (if (= 1 (modulo n 3)) 1 0) (A292595 (A285712 n)))))

Formula

a(1) = 0, a(2) = 1, and for n > 1, a(n) = a(A285712(n)) + [1 == (n mod 3)].
a(n) = A000120(A292591(n)).
a(n) + A292594(n) = A285715(n).

A292244 Base-2 expansion of a(n) encodes the steps where multiples of 3 are encountered when map x -> A253889(x) is iterated down to 1, starting from x=n.

Original entry on oeis.org

0, 0, 1, 0, 0, 3, 0, 2, 5, 0, 0, 1, 0, 0, 1, 12, 6, 7, 14, 0, 1, 0, 4, 1, 8, 10, 3, 0, 0, 21, 24, 0, 1, 28, 2, 3, 2, 0, 1, 0, 0, 5, 2, 2, 1, 22, 24, 17, 0, 12, 33, 32, 14, 35, 42, 28, 45, 24, 0, 1, 16, 2, 11, 48, 0, 59, 0, 8, 3, 0, 2, 5, 0, 16, 1, 4, 20, 3, 6, 6, 7, 8, 0, 1, 56, 0, 3, 0, 42, 5, 0, 48, 5, 0, 0, 1, 14, 2, 65, 64, 56, 49, 44, 4, 49, 64, 6, 57, 0
Offset: 1

Views

Author

Antti Karttunen, Sep 15 2017

Keywords

Examples

			For n = 3, the starting value is a multiple of three, after which follows A253889(3) = 1, the end point of iteration, which is not a multiple of three, thus a(3) = 1*(2^0) = 1.
For n = 8, the starting value is not a multiple of three, after which follows A253889(8) = 3, which is, thus a(8) = 0*(2^0) + 1*(2^1) = 2.
For n = 9, the starting value is a multiple of three, after which follows A253889(9) = 8 (which is not), while A253889(8) = 3 (which is), thus a(9) = 1*(2^0) + 0*(2^1) + 1*(2^2) = 5.

Links

Crossrefs

Cf. A048673, A064216, A253889, A291770, A292243, A292247.
Cf. also A292245, A292246, and A292381, A292383, A292385, and A292590, A292591 for similarly constructed sequences, and also A292250.

Programs

  • Mathematica
    f[n_] := Times @@ Power[If[# == 1, 1, NextPrime[#, -1]] & /@ First@ #, Last@ #] &@ Transpose@ FactorInteger[2 n - 1];g[n_] := (Times @@ Power[If[# == 1, 1, NextPrime@ #] & /@ First@ #, Last@ #] + 1)/2 &@ Transpose@ FactorInteger@ n;Table[FromDigits[#, 2] &@ Map[Boole[Divisible[#, 3]] &,  Reverse@ NestWhileList[Floor@ g[Floor[f[#]/2]] &, n, # > 1 &]], {n, 109}] (* Michael De Vlieger, Sep 16 2017 *)
  • Scheme
    (define (A292244 n) (A291770 (A292243 n)))

Formula

a(n) = A291770(A292243(n)).
Other identities. For all n >= 1:
a(A048673(n)) = A292247(n).
a(n) + A292245(n) = A064216(n).
a(n) AND A292245(n) = a(n) AND A292246(n) = 0, where AND is a bitwise-AND (A004198).

A292590 a(1) = 0; and for n > 1, a(n) = 2*a(A285712(n)) + [0 == (n mod 3)].

Original entry on oeis.org

0, 0, 1, 2, 0, 5, 10, 2, 21, 42, 4, 85, 0, 0, 171, 342, 10, 5, 684, 20, 1369, 2738, 4, 5477, 0, 42, 10955, 8, 84, 21911, 43822, 8, 21, 87644, 170, 175289, 350578, 0, 11, 701156, 0, 1402313, 40, 342, 2804627, 16, 684, 85, 5609254, 20, 11218509, 22437018, 10, 44874037, 89748074, 1368, 179496149, 168, 40, 43, 0, 2738, 1, 358992298, 5476, 717984597, 80, 8
Offset: 1

Views

Author

Antti Karttunen, Sep 20 2017

Keywords

Comments

Binary expansion of a(n) encodes the positions of multiples of three in the path taken from n to the root in the binary trees like A245612 and A244154.

Links

Crossrefs

Cf. A079978, A244153, A245611, A285712, A292591, A292592, A292594.
Cf. also A292244, A292383.

Programs

  • Mathematica
    f[n_] := f[n] = Which[n == 1, 0, Mod[n, 3] == 2, Ceiling[n/3], True, (Times @@ Power[If[# == 1, 1, NextPrime[#, -1]] & /@ First@ #, Last@ #] &@ Transpose@ FactorInteger[2 n - 1] + 1)/2]; a[n_] := a[n] = If[n == 1, n - 1, 2 a[f@ n] + Boole[Divisible[n, 3]]]; Array[a, 67] (* Michael De Vlieger, Sep 22 2017 *)
  • Scheme
    (define (A292590 n) (if (<= n 1) 0 (+ (if (zero? (modulo n 3)) 1 0) (* 2 (A292590 (A285712 n))))))

Formula

a(1) = 0; and for n > 1, a(n) = A079978(n) + 2*a(A285712(n)).
a(n) + A292591(n) = A245611(n).
a(A245612(n)) = A292592(n).
A000120(a(n)) = A292594(n).
Showing 1-4 of 4 results.

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