A261891 -id:A261891 - cp's OEIS frontend

A261892 Least multiple m of n such that n and m have no common one bit in their binary representations.

2, 4, 12, 8, 10, 24, 56, 16, 18, 20, 132, 48, 130, 112, 240, 32, 34, 36, 76, 40, 42, 264, 552, 96, 100, 260, 324, 224, 290, 480, 992, 64, 66, 68, 140, 72, 74, 152, 1560, 80, 82, 84, 516, 528, 450, 1104, 2256, 192, 196, 200, 204, 520, 1802, 648, 3080, 448

Offset: 1

Paul Tek, Sep 05 2015

All terms are even.
a(A003714(n)) = 2*A003714(n) for any n>0.
a(2n) = 2*a(n) for any n>0.
Without the condition on m being a multiple of n, then we'd get another sequence b(n) that seems to be A006519(n+1). - Michel Marcus, Sep 06 2015

			For n=7:
+---+-----+---------------+-----------------+
| k | 7*k | 7*k in binary | Common one bits |
+---+-----+---------------+-----------------+
| 1 |   7 |           111 |             111 |
| 2 |  14 |          1110 |             110 |
| 3 |  21 |         10101 |             101 |
| 4 |  28 |         11100 |             100 |
| 5 |  35 |        100011 |              11 |
| 6 |  42 |        101010 |              10 |
| 7 |  49 |        110001 |               1 |
| 8 |  56 |        111000 |               0 |
+---+-----+---------------+-----------------+
Hence, a(7) = 56.

Paul Tek, Table of n, a(n) for n = 1..16384

Cf. A003714, A261891.

Table[k = 1; While[BitAnd[k n, n] != 0, k++]; k n, {n, 60}] (* Michael De Vlieger, Sep 06 2015 *)

a(n) = {k=1; while (bitand(n, k*n), k++); k*n;} \\ Michel Marcus, Sep 06 2015

sub a {
    my $n = shift;
    my $m = $n;
    while ($n & $m) {
        $m += $n;
    }
    return $m;
}

from itertools import count
def A261892(n): return next(k for k in count(n<<1,n) if not n&k) # Chai Wah Wu, Jul 19 2024

a(n) = n*A261891(n) for any n>0.

A353623 a(n) is the least k > 0 such that n and k*n can be added without carries in balanced ternary.

Original entry on oeis.org

1, 2, 3, 2, 2, 6, 3, 3, 2, 2, 2, 5, 2, 2, 6, 6, 5, 3, 3, 3, 3, 3, 6, 2, 2, 3, 2, 2, 2, 3, 2, 2, 9, 5, 12, 2, 2, 2, 12, 2, 2, 6, 6, 12, 11, 6, 6, 14, 5, 9, 8, 3, 3, 3, 3, 3, 3, 3, 9, 11, 3, 3, 3, 3, 3, 14, 6, 6, 2, 2, 9, 2, 2, 2, 3, 3, 6, 2, 2, 3, 2, 2, 2, 3, 2

Offset: 0

Rémy Sigrist, Apr 30 2022

Two integers can be added without carries in balanced ternary if they have no equal nonzero digit at the same position.

			For n = 5:
- we consider the following cases:
      k  bter(k*5)  carries?
      -  ---------  --------
      1        1TT  yes
      2        101  yes
      3       1TT0  yes
      4       1T1T  yes
      5       10T1  yes
      6       1010  no
- so a(5) = 6.

Rémy Sigrist, Table of n, a(n) for n = 0..10000
Wikipedia, Balanced ternary

Cf. A059095, A261891 (binary analog), A353624.

ok(u,v) = { while (u && v, my (uu=[0,+1,-1][1+u%3], vv=[0,+1,-1][1+v%3]); if (abs(uu+vv)>1, return (0)); u=(u-uu)/3; v=(v-vv)/3); return (1) }
a(n) = for (k=1, oo, if (ok(n, n*k), return (k)))

a(n) = A353624(n) / n for any n > 0.

a(3*n) = a(n).

A331985 a(n) is the least positive k such that n AND floor(n/k) = 0 (where AND denotes the bitwise AND operator).

Original entry on oeis.org

1, 2, 2, 4, 2, 2, 4, 8, 2, 2, 2, 12, 4, 5, 8, 16, 2, 2, 2, 4, 2, 2, 12, 24, 4, 4, 5, 6, 8, 10, 16, 32, 2, 2, 2, 4, 2, 2, 4, 40, 2, 2, 2, 9, 12, 16, 24, 48, 4, 4, 4, 4, 5, 5, 6, 56, 8, 9, 10, 12, 16, 21, 32, 64, 2, 2, 2, 4, 2, 2, 4, 8, 2, 2, 2, 16, 4, 26, 40

Offset: 0

Rémy Sigrist, Feb 03 2020

This sequence has similarities with A261891; here we divide and round down, there we multiply, in order to obtain a number with no common bit with the original.

			For n = 3:
- 3 AND floor(3/1) = 3,
- 3 AND floor(3/2) = 1,
- 3 AND floor(3/3) = 1,
- 3 AND floor(3/4) = 0,
- hence a(3) = 4.

Rémy Sigrist, Table of n, a(n) for n = 0..8192

Cf. A003714, A261891, A332011.

a(n) = for (k=1, oo, if (bitand(n,n\k)==0, return (k)))

a(n) = 2 iff n is a positive Fibbinary number (A003714).

A374735 a(n) is the least k > 0 such that n and k*n can be added without carries in decimal.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 3, 5, 10, 1, 1, 1, 1, 1, 2, 2, 3, 10, 20, 1, 1, 1, 1, 1, 2, 2, 6, 5, 30, 1, 1, 1, 1, 1, 4, 7, 3, 20, 40, 1, 1, 1, 1, 1, 10, 5, 3, 5, 50, 2, 2, 2, 2, 6, 2, 2, 6, 30, 60, 2, 2, 2, 2, 5, 2, 2, 3, 15, 70, 3, 3, 3, 7, 3, 4, 12, 13, 40, 80, 5, 5

Offset: 0

Rémy Sigrist, Jul 18 2024

			For n = 8:
- 1*8 = 8; computing 8 + 8 requires a carry,
- 2*8 = 16; computing 8 + 16 requires a carry,
- 3*8 = 24; computing 8 + 24 requires a carry,
- 4*8 = 32; computing 8 + 32 requires a carry,
- 5*8 = 40; computing 8 + 40 does not require a carry,
- so a(8) = 5.

Rémy Sigrist, Table of n, a(n) for n = 0..10000

Cf. A007091, A261891 (analog for binary), A353623 (analog for balanced ternary), A374736.

a(n, base = 10) = { for (k = 1, oo, if (sumdigits((k+1)*n, base) == sumdigits(n, base) + sumdigits(k*n, base), return (k););); }

from itertools import count
def A374735(n):
    s = list(map(int,str(n)[::-1]))
    return next(k for k in count(1) if all(a+b<=9 for a, b in zip(s,map(int,str(k*n)[::-1])))) # Chai Wah Wu, Jul 19 2024

a(n) = 1 iff n belongs to A007091.

a(10*n) = a(n).

cp's OEIS Frontend

A261892 Least multiple m of n such that n and m have no common one bit in their binary representations.

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A353623 a(n) is the least k > 0 such that n and k*n can be added without carries in balanced ternary.

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A331985 a(n) is the least positive k such that n AND floor(n/k) = 0 (where AND denotes the bitwise AND operator).

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A374735 a(n) is the least k > 0 such that n and k*n can be added without carries in decimal.

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