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.

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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).

A336121 a(1) = 0, and for n > 1, a(n) = [A122111(n) == 3 (mod 4)] + a(A253553(n)).

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 2, 0, 2, 0, 1, 0, 1, 0, 2, 1, 1, 0, 1, 0, 2, 0, 3, 0, 1, 1, 1, 0, 1, 0, 2, 0, 2, 0, 1, 0, 1, 0, 3, 0, 1, 0, 1, 0, 3, 1, 2, 0, 1, 0, 1, 0, 1, 0, 3, 1, 2, 0, 1, 0, 1, 0, 2, 0, 1, 1, 1, 0, 2, 0, 3, 0, 1, 0, 1, 1, 1, 0, 2, 0, 3, 0, 1, 0, 1, 1, 3, 0, 2, 0, 2, 0, 2, 0, 2, 1
Offset: 1

Views

Author

Antti Karttunen, Jul 17 2020

Keywords

Comments

Positions for the first occurrence of each n, for n >= 0, are: 1, 4, 16, 32, 144, 512, 2048, 6912, 20736, 62208, ...

Links

Crossrefs

Cf. A000120, A001222, A122111, A253553, A292377, A292383, A336120, A336123, A336124.
Cf. A336119 (positions of zeros).

Programs

  • PARI
    A253553(n) = if(n<=2,1,my(f=factor(n), k=#f~); if(f[k,2]>1,f[k,2]--,f[k,1] = precprime(f[k,1]-1)); factorback(f));
A336121(n) = if(1==n,0,(3==A336124(n))+A336121(A253553(n)));

Formula

a(1) = 0, and for n > 1, a(n) = [A336124(n) == 3] + a(A253553(n)).
a(n) = A000120(A336120(n)).
a(n) = A292377(A122111(n)).
a(n) = A001222(n) - A336123(n).

A292603 Doudna-tree reduced modulo 4: a(n) = A005940(1+n) mod 4.

Original entry on oeis.org

1, 2, 3, 0, 1, 2, 1, 0, 3, 2, 3, 0, 1, 2, 3, 0, 3, 2, 1, 0, 3, 2, 1, 0, 1, 2, 3, 0, 1, 2, 1, 0, 1, 2, 1, 0, 3, 2, 3, 0, 1, 2, 1, 0, 3, 2, 3, 0, 1, 2, 3, 0, 1, 2, 1, 0, 3, 2, 3, 0, 1, 2, 3, 0, 1, 2, 3, 0, 1, 2, 3, 0, 3, 2, 1, 0, 3, 2, 1, 0, 3, 2, 3, 0, 1, 2, 3, 0, 3, 2, 1, 0, 3, 2, 1, 0, 1, 2, 3, 0, 1, 2, 1, 0, 3, 2, 3, 0, 1, 2, 3, 0, 3, 2, 1, 0, 3, 2, 1, 0, 1
Offset: 0

Views

Author

Antti Karttunen, Dec 01 2017

Keywords

Examples

			The first six levels of the binary tree (compare also to the illustrations given at A005940 and A292602):
                               1
                               |
                               2
                ............../ \..............
               3                               0
        ....../ \......                 ....../ \......
       1               2               1               0
      / \             / \             / \             / \
     /   \           /   \           /   \           /   \
    3     2         3     0         1     2         3     0
   / \   / \       / \   / \       / \   / \       / \   / \
  3   2 1   0     3   2 1   0     1   2 3   0     1   2 1   0

Links

Crossrefs

Cf. A003961, A005940, A292602.
Cf. A004767 (gives the positions of 0's), A016813 (of 2's).
Cf. also A246261, A246263, A246271, A292271, A292274, A292375, A292377, A292381, A292383, A292384, A292583.

Programs

Formula

a(n) = A010873(A005940(1+n)).
a(n) + 4*A292602(n) = A005940(1+n).
a(2n+1) = 2*a(n) mod 4.
a(A004767(n)) = 0.
a(A016813(n)) = 2.
a(2*A156552(A246261(n))) = 1.
a(2*A156552(A246263(n))) = 3.
a(n * 2^(1+A246271(A005940(1+n)))) = 1.

A332900 Lexicographically earliest infinite sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = 0 if n > 1 and n is a square or twice square, with f(n) = n for all other numbers.

Original entry on oeis.org

1, 2, 3, 2, 4, 5, 6, 2, 2, 7, 8, 9, 10, 11, 12, 2, 13, 2, 14, 15, 16, 17, 18, 19, 2, 20, 21, 22, 23, 24, 25, 2, 26, 27, 28, 2, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 2, 2, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 2, 54, 55, 56, 57, 58, 59, 60, 2, 61, 62, 63, 64, 65, 66, 67, 68, 2, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 2, 85, 2
Offset: 1

Views

Author

Antti Karttunen, Mar 05 2020

Keywords

Comments

For all i, j:
A324400(i) = A324400(j) => a(i) = a(j),
a(i) = a(j) => A292383(i) = A292383(j) => A292583(i) = A292583(j),
a(i) = a(j) => A332896(i) = A332896(j) => A332901(i) = A332901(j).

Links

Crossrefs

Cf. A292383, A292583, A324400, A332896, A332901.

Programs

  • PARI
    up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A332900aux(n) = if((n>1)&&(issquare(n)||issquare(2*n)),0,n);
v332900 = rgs_transform(vector(up_to,n,A332900aux(n)));
A332900(n) = v332900[n];
Previous Showing 21-24 of 24 results.

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