A175488 -id:A175488 - cp's OEIS frontend

A175466 Table read by antidiagonals: a(m,n) = the largest positive integer occurring, when written in binary, as a substring in both binary m and binary n.

1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 3, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 4, 1, 2, 1, 1, 1, 3, 2, 2, 3, 1, 1, 1, 2, 3, 2, 5, 2, 3, 2, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 1, 1, 2, 1, 4, 1, 6, 1, 4, 1, 2, 1, 1, 2, 1, 4, 2, 3, 3, 2, 4, 1, 2, 1, 1, 2, 3, 2, 2, 2, 7, 2, 2

Offset: 1

Leroy Quet, May 24 2010

			From _Rémy Sigrist_, Jul 20 2019: (Start)
Table a(n, k) begins (in decimal):
  n\k|  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15
  ---+---------------------------------------------------
    1|  1  1  1  1  1  1  1  1  1   1   1   1   1   1   1
    2|  1  2  1  2  2  2  1  2  2   2   2   2   2   2   1
    3|  1  1  3  1  1  3  3  1  1   1   3   3   3   3   3
    4|  1  2  1  4  2  2  1  4  4   2   2   4   2   2   1
    5|  1  2  1  2  5  2  1  2  2   5   5   2   5   2   1
    6|  1  2  3  2  2  6  3  2  2   2   3   6   6   6   3
    7|  1  1  3  1  1  3  7  1  1   1   3   3   3   7   7
    8|  1  2  1  4  2  2  1  8  4   2   2   4   2   2   1
    9|  1  2  1  4  2  2  1  4  9   2   2   4   2   2   1
   10|  1  2  1  2  5  2  1  2  2  10   5   2   5   2   1
   11|  1  2  3  2  5  3  3  2  2   5  11   3   5   3   3
   12|  1  2  3  4  2  6  3  4  4   2   3  12   6   6   3
   13|  1  2  3  2  5  6  3  2  2   5   5   6  13   6   3
   14|  1  2  3  2  2  6  7  2  2   2   3   6   6  14   7
   15|  1  1  3  1  1  3  7  1  1   1   3   3   3   7  15
Table a(n, k) begins (in binary):
   n\k| 1 10 11 100 101 110 111 1000 1001 1010 1011 1100 1101 1110 1111
  ----+----------------------------------------------------------------
     1| 1  1  1   1   1   1   1    1    1    1    1    1    1    1    1
    10| 1 10  1  10  10  10   1   10   10   10   10   10   10   10    1
    11| 1  1 11   1   1  11  11    1    1    1   11   11   11   11   11
   100| 1 10  1 100  10  10   1  100  100   10   10  100   10   10    1
   101| 1 10  1  10 101  10   1   10   10  101  101   10  101   10    1
   110| 1 10 11  10  10 110  11   10   10   10   11  110  110  110   11
   111| 1  1 11   1   1  11 111    1    1    1   11   11   11  111  111
  1000| 1 10  1 100  10  10   1 1000  100   10   10  100   10   10    1
  1001| 1 10  1 100  10  10   1  100 1001   10   10  100   10   10    1
  1010| 1 10  1  10 101  10   1   10   10 1010  101   10  101   10    1
  1011| 1 10 11  10 101  11  11   10   10  101 1011   11  101   11   11
  1100| 1 10 11 100  10 110  11  100  100   10   11 1100  110  110   11
  1101| 1 10 11  10 101 110  11   10   10  101  101  110 1101  110   11
  1110| 1 10 11  10  10 110 111   10   10   10   11  110  110 1110  111
  1111| 1  1 11   1   1  11 111    1    1    1   11   11   11  111 1111
(End)

Rémy Sigrist, Table of n, a(n) for n = 1..11325
Rémy Sigrist, Colored representation of the table for n = 1..1023 and k = 1023

Cf. A043529, A004780, A175488.

sub(n) = { my (b=binary(n), s=[]); for (i=1, #b, if (b[i], for (j=i, #b, s=setunion(s, Set(fromdigits(b[i..j], 2)))))); return (s) }
T(n,k) = my (i=setintersect(sub(n), sub(k))); i[#i] \\ Rémy Sigrist, Jul 20 2019

From Rémy Sigrist, Jul 20 2019: (Start)
1 <= a(n, k) <= min(n, k).
a(n, k) = a(k, n).
a(n, n) = n.
a(n, 1) = 1.
a(n, 2) = A043529(n).
a(n, 3) = 3 iff n belongs to A004780. (End)

More terms from Rémy Sigrist, Jul 20 2019

A175489 a(n) = Sum_{k=1..n} b(k,n), where b(k,n) is the largest positive integer that, when written in binary, occurs as a substring in both binary k and binary n.

Original entry on oeis.org

1, 3, 5, 8, 11, 16, 17, 21, 26, 28, 39, 46, 55, 58, 51, 54, 63, 65, 85, 87, 82, 104, 118, 128, 144, 147, 163, 176, 193, 181, 141, 137, 154, 156, 192, 162, 212, 231, 252, 247, 275, 202, 267, 307, 306, 333, 336, 345, 373, 381, 410, 412, 419, 423, 475, 497, 532, 539

Offset: 1

Leroy Quet, May 28 2010

Row sums of A175488. - R. J. Mathar, Sep 28 2010

Rémy Sigrist, Table of n, a(n) for n = 1..8192
Rémy Sigrist, PARI program for A175489

Cf. A175488, A175490.

A175488 := proc(n,m) for a from m to 1 by -1 do abin := convert(a,base,2) ; nbin := convert(n,base,2) ; mbin := convert(m,base,2) ; if verify(abin,nbin,'sublist') and verify(abin,mbin,'sublist') then return a; end if; end do: end proc:
A175489 := proc(n) add(A175488(n,m),m=1..n) ; end proc:
seq(A175489(n),n=1..50) ;
# R. J. Mathar, Sep 28 2010

PARI
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See Links section.
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More terms from R. J. Mathar, Sep 28 2010

A175490 a(n) = Product_{k=1..n} b(k,n), where b(k,n) is the largest positive integer that, when written in binary, occurs as a substring in both binary k and binary n.

Original entry on oeis.org

1, 2, 3, 8, 20, 144, 63, 256, 1152, 1600, 118800, 995328, 8424000, 12192768, 178605, 1048576, 8912896, 21233664, 27572576256, 13107200000, 537600000, 2032335360000, 30928035600000, 760840571584512, 13541327555788800

Offset: 1

Leroy Quet, May 28 2010

Row products of A175488. - R. J. Mathar, Sep 28 2010

Rémy Sigrist, Table of n, a(n) for n = 1..819
Rémy Sigrist, PARI program for A175490

Cf. A175488, A175489.

A175488 := proc(n,m) for a from m to 1 by -1 do abin := convert(a,base,2) ; nbin := convert(n,base,2) ; mbin := convert(m,base,2) ; if verify(abin,nbin,'sublist') and verify(abin,mbin,'sublist') then return a; end if; end do: end proc:
A175490 := proc(n) mul(A175488(n,m),m=1..n) ; end proc:
seq(A175490(n),n=1..30) ;
# R. J. Mathar, Sep 28 2010

PARI
```
See Links section.
```

More terms from R. J. Mathar, Sep 28 2010

cp's OEIS Frontend

A175466 Table read by antidiagonals: a(m,n) = the largest positive integer occurring, when written in binary, as a substring in both binary m and binary n.

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A175489 a(n) = Sum_{k=1..n} b(k,n), where b(k,n) is the largest positive integer that, when written in binary, occurs as a substring in both binary k and binary n.

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A175490 a(n) = Product_{k=1..n} b(k,n), where b(k,n) is the largest positive integer that, when written in binary, occurs as a substring in both binary k and binary n.

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