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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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

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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.

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.

A065160 Reduced binary string self-substitutions: a(n) is obtained by substituting n for each 1-bit in the binary expansion of n, then dividing by n.

Original entry on oeis.org

1, 2, 5, 4, 17, 18, 73, 8, 65, 66, 529, 68, 545, 546, 4369, 16, 257, 258, 4129, 260, 4161, 4162, 66593, 264, 4225, 4226, 67617, 4228, 67649, 67650, 1082401, 32, 1025, 1026, 32833, 1028, 32897, 32898, 1052737, 1032, 33025, 33026, 1056833, 33028, 1056897
Offset: 1

Views

Author

Marc LeBrun, Oct 18 2001

Keywords

Comments

By convention a(0)=0. a(2^n)=2^n. a(4n+2)=1+a(4n+1). a(n)=A065158(n,n)=A065157(n,n)/n=A065159(n)/n.

Examples

			a(5): 5=101->(101)0(101)=1010101=85; 85/5=17.

Crossrefs

Cf. A065157, A065158. Equals A065159(n)/n.

Formula

a(2^n)=2^n. a(4n+2)=1+a(4n+1). a(n)=A065158(n, n)=A065157(n, n)/n=A065159(n)/n.
a(n)=z(n, A062383(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)*v+1. - Reinhard Zumkeller, Feb 15 2004
Showing 1-3 of 3 results.

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