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 1-3 of 3 results.

A065157 Table of binary string substitutions: a(i,j) is obtained by substituting i for each 1-bit in j.

Original entry on oeis.org

1, 2, 2, 3, 4, 3, 4, 6, 10, 4, 5, 8, 15, 8, 5, 6, 10, 36, 12, 18, 6, 7, 12, 45, 16, 27, 20, 7, 8, 14, 54, 20, 68, 30, 42, 8, 9, 16, 63, 24, 85, 72, 63, 16, 9, 10, 18, 136, 28, 102, 90, 292, 24, 34, 10, 11, 20, 153, 32, 119, 108, 365, 32, 51, 36, 11, 12, 22, 170, 36, 264, 126, 438, 40, 132, 54, 74, 12
Offset: 1

Views

Author

Marc LeBrun, Oct 18 2001

Keywords

Examples

			a(3,5): 5 = 101_2 -> (3)0(3) = (11)0(11)_2 = 11011_2 = 27.
a(5,3): 3 = 11_2 -> (5)(5) = (101)(101)_2 = 101101_2 = 45.

Crossrefs

Cf. A065158, A065159, A065160.

Programs

  • PARI
    T(n,k) = my(bk=binary(k), sn=Str(fromdigits(binary(n))), s=""); for (i=1, #bk, if (bk[i] == 1, s=concat(s, sn), s=concat(s, "0"))); fromdigits(apply(eval, Vec(s)), 2); \\ Michel Marcus, Feb 11 2023

Formula

Table origin is a(1,1).
a(1,n) = a(n,1) = n.
a(0,n) = a(n,0) = 0.
a(i,j) = A065158(i,j)*i.
a(n,n) = A065159(n) = A065160(n)*n.

A065159 Binary string self-substitutions: a(n) is obtained by substituting the binary expansion of n for each 1-bit in the binary expansion of n.

Original entry on oeis.org

0, 1, 4, 15, 16, 85, 108, 511, 64, 585, 660, 5819, 816, 7085, 7644, 65535, 256, 4369, 4644, 78451, 5200, 87381, 91564, 1531639, 6336, 105625, 109876, 1825659, 118384, 1961821, 2029500, 33554431, 1024, 33825, 34884, 1149155, 37008, 1217189, 1250124, 41056743
Offset: 0

Views

Author

Marc LeBrun, Oct 18 2001

Keywords

Examples

			a(5): 5 = 101 -> (101)0(101) = 1010101 = 85.

Links

Crossrefs

Cf. A007088, A065157, A065158, A065160.

Programs

  • Mathematica
    bss[n_]:=Module[{idn2=IntegerDigits[n,2]},FromDigits[Flatten[idn2/.{1-> idn2}],2]]; Array[bss,40,0] (* Harvey P. Dale, Aug 15 2017 *)
  • Python
    def a(n): b = bin(n)[2:]; return int(b.replace("1", b), 2)
print([a(n) for n in range(40)]) # Michael S. Branicky, Aug 05 2022

Formula

a(0) = 0. a(2^n) = 4^n. a(4n+2) = (4n+2)*(1+a(4n+1)/(4n+1)).
a(n) = A065157(n,n) = A065158(n,n)*n = A065160(n)*n.
a(n) =z(n, n) with z(u, v) = if u=0 then 0 else if u mod 2 = 0 then z(u/2, v)*2 else z([u/2], v)*A062383(v)+v. - Reinhard Zumkeller, Feb 15 2004

Extensions

Name clarified by Michael S. Branicky, Aug 05 2022

A065158 Table of reduced binary string substitutions: a(i,j) is obtained by substituting i for each 1-bit in j, then dividing by i.

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 1, 2, 5, 4, 1, 2, 5, 4, 5, 1, 2, 9, 4, 9, 6, 1, 2, 9, 4, 9, 10, 7, 1, 2, 9, 4, 17, 10, 21, 8, 1, 2, 9, 4, 17, 18, 21, 8, 9, 1, 2, 17, 4, 17, 18, 73, 8, 17, 10, 1, 2, 17, 4, 17, 18, 73, 8, 17, 18, 11, 1, 2, 17, 4, 33, 18, 73, 8, 33, 18, 37, 12, 1, 2, 17, 4, 33, 34, 73, 8, 33
Offset: 1

Views

Author

Marc LeBrun, Oct 18 2001

Keywords

Comments

Table origin is a(1,1). a(1,n)=n. a(n,1)=1. By convention a(0,n)=a(n,0)=0. a(i,j)=A065157(i,j)/i. a(n,n)=A065160(n)=A065159(n)/n.

Examples

			a(3,5): 5=101->(3)0(3)=(11)0(11)=11011=27; 27/3=9. a(5,3): 3=11->(5)(5)=(101)(101)=101101=45; 45/5=9

Crossrefs

A065157, A065159, A065160.
Showing 1-3 of 3 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.