A261892 -id:A261892 - cp's OEIS frontend

A261891 Least k>0 such that n AND (k*n) = 0, where AND stands for the binary AND operator.

2, 2, 4, 2, 2, 4, 8, 2, 2, 2, 12, 4, 10, 8, 16, 2, 2, 2, 4, 2, 2, 12, 24, 4, 4, 10, 12, 8, 10, 16, 32, 2, 2, 2, 4, 2, 2, 4, 40, 2, 2, 2, 12, 12, 10, 24, 48, 4, 4, 4, 4, 10, 34, 12, 56, 8, 18, 10, 12, 16, 42, 32, 64, 2, 2, 2, 4, 2, 2, 4, 8, 2, 2, 2, 12, 4, 10

Offset: 1

Paul Tek, Sep 05 2015

All terms are even.
a(A003714(n)) = 2 for any n>0.
a(A004780(n)) > 2 for any n>0.
a(n) <= 2^A116361(n) for any n>0.
a(2n) = a(n) for any n>0.

			For n=7:
+---+-------------+
| k | 7 AND (k*7) |
|   | (in binary) |
+---+-------------+
| 1 |         111 |
| 2 |         110 |
| 3 |         101 |
| 4 |         100 |
| 5 |          11 |
| 6 |          10 |
| 7 |           1 |
| 8 |           0 |
+---+-------------+
Hence, a(7) = 8.

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

Cf. A003714, A004780, A116361, A261892.

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

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

sub a {
   my $n = shift;
   my $k = 1;
   while ($n & ($k*$n)) {
      $k++;
   }
   return $k;
}

from itertools import count
def A261891(n): return next(k for k in count(2) if not n&k*n) # Chai Wah Wu, Jul 19 2024

A353624 a(0) = 0, and for n > 0, a(n) is the least multiple of n that can be added to n without carries in balanced ternary.

Original entry on oeis.org

0, 2, 6, 6, 8, 30, 18, 21, 16, 18, 20, 55, 24, 26, 84, 90, 80, 51, 54, 57, 60, 63, 132, 46, 48, 75, 52, 54, 56, 87, 60, 62, 288, 165, 408, 70, 72, 74, 456, 78, 80, 246, 252, 516, 484, 270, 276, 658, 240, 441, 400, 153, 156, 159, 162, 165, 168, 171, 522, 649

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*5 = 30.

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

Cf. A059095, A261892 (binary analog), A353623.

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 (n*k)))

a(n) = n * A353623(n).

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

A374736 a(n) is the least number of the form k*n for some k > 0 that can be added to n without carries in decimal.

Original entry on oeis.org

0, 1, 2, 3, 4, 10, 12, 21, 40, 90, 10, 11, 12, 13, 14, 30, 32, 51, 180, 380, 20, 21, 22, 23, 24, 50, 52, 162, 140, 870, 30, 31, 32, 33, 34, 140, 252, 111, 760, 1560, 40, 41, 42, 43, 44, 450, 230, 141, 240, 2450, 100, 102, 104, 106, 324, 110, 112, 342, 1740

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) = 40.

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

Cf. A007091, A261892 (analog for binary), A353624 (analog for balanced ternary), A374735.

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

from itertools import count
def A374736(n):
    s = list(map(int,str(n)[::-1]))
    return next(k for k in count(n,n) if all(a+b<=9 for a, b in zip(s,map(int,str(k)[::-1])))) # Chai Wah Wu, Jul 19 2024

a(n) = A374735(n) * n.
a(n) = n iff n belongs to A007091.
a(10*n) = 10*a(n).

cp's OEIS Frontend

A261891 Least k>0 such that n AND (k*n) = 0, where AND stands for the binary AND operator.

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A353624 a(0) = 0, and for n > 0, a(n) is the least multiple of n that can be added to n without carries in balanced ternary.

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A374736 a(n) is the least number of the form k*n for some k > 0 that can be added to n without carries in decimal.

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