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.

A292251 The 3-adic valuation of A048673(n).

Original entry on oeis.org

0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 2, 0, 2, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 3, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 2, 0, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 1, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 4, 0, 1, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0
Offset: 1

Views

Author

Antti Karttunen, Sep 12 2017

Keywords

Links

Crossrefs

One less than the even bisection of A292252.
Cf. A007814, A007949, A048673, A292250.
Cf. also A292241, A292261.

Programs

  • Mathematica
    IntegerExponent[#, 3] & /@ Table[(Times @@ Power[If[# == 1, 1, NextPrime@ #] & /@ First@ #, Last@ #] + 1)/2 &@ Transpose@ FactorInteger@ n, {n, 120}] (* Michael De Vlieger, Sep 12 2017 *)

Formula

a(n) = A007814(1+A292250(n)).
a(n) = A007949(A048673(n)).
a(n) = A007949(3*A048673(n)) - 1.
a(n) = A292252(2n)-1.

A304759 Binary encoding of 1-digits in ternary representation of A048673(n).

Original entry on oeis.org

1, 0, 2, 2, 3, 0, 0, 6, 7, 4, 1, 2, 4, 4, 0, 14, 5, 12, 6, 10, 9, 0, 4, 6, 1, 0, 4, 10, 5, 8, 1, 30, 8, 8, 14, 26, 2, 8, 13, 22, 3, 16, 0, 2, 17, 12, 8, 14, 1, 0, 10, 2, 10, 0, 9, 22, 3, 8, 11, 18, 9, 0, 18, 62, 0, 20, 12, 18, 1, 24, 13, 54, 15, 0, 28, 18, 0, 24, 12, 46, 37, 4, 8, 34, 7, 4, 0, 6, 11, 32, 23, 26, 22, 0
Offset: 1

Views

Author

Antti Karttunen, May 30 2018

Keywords

Comments

Compare the logarithmic scatterplot to those of A291759, A292250 and A304760.

Links

Crossrefs

Cf. A048673, A289813, A304758 (rgs-transform), A340381.
Cf. also A291759, A292250, A304760, A305295.
Cf. A340376 (positions of zeros), A340378 (binary weight).

Programs

Formula

a(n) = A289813(A048673(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).

A292240 Binary encoding of 0-digits in ternary representation of A254103(n).

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Sep 12 2017

Keywords

Links

Crossrefs

Cf. A254103, A291760, A291770, A292241, A292242.
Cf. also A292250, A292260.

Formula

a(n) = A291770(A254103(n)).

A341345 a(n) = A048673(n) mod 3.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Feb 09 2021

Keywords

Links

Crossrefs

Cf. A003961, A010872, A048673, A292247, A292248, A292251, A292252.
Cf. A007395 (even bisection), A341346 (odd bisection), A341347.
Cf. also A291759, A292243, A292244, A292245, A292246, A292250, A304759, A340376, A340377.
Cf. also A292603.

Programs

  • PARI
    A003961(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); }; \\ From A003961
A341345(n) = (((A003961(n)+1)/2)%3);

Formula

a(n) = A010872(A048673(n)).
a(n) = 0 iff A292247(n) is odd.
a(n) = 0 iff A292250(n) is odd, or equally, iff both A291759(n) and A304759(n) are even.
a(n) = 0 iff A292251(n) > 0.
a(n) = 1 iff A292248(n) is odd.
a(n) = 1 iff A304759(n) is odd, or equally, iff both A291759(n) and A292250(n) are even.
a(2n) = 2.
Showing 1-5 of 5 results.

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