A328178 a(n) is the minimal value of the expression d XOR (n/d) where d runs through the divisors of n and XOR denotes the bitwise XOR operator.

0, 3, 2, 0, 4, 1, 6, 6, 0, 7, 10, 4, 12, 5, 6, 0, 16, 5, 18, 1, 4, 9, 22, 2, 0, 15, 10, 3, 28, 3, 30, 12, 8, 19, 2, 0, 36, 17, 14, 13, 40, 1, 42, 15, 12, 21, 46, 8, 0, 15, 18, 9, 52, 15, 14, 10, 16, 31, 58, 9, 60, 29, 14, 0, 8, 13, 66, 21, 20, 11, 70, 1, 72

Offset

1,2

Links

Rémy Sigrist, Table of n, a(n) for n = 1..16384

Rémy Sigrist, Logarithmic scatterplot of (n, 1+a(n)) for n = 1..2^16

Formula

a(n) = 0 iff n is a square.

a(p) = p-1 for any odd prime number p.

Example

For n = 12:
- we have the following values:
    d   12/d  d XOR (12/d)
    --  ----  ------------
     1    12            13
     2     6             4
     3     4             7
     4     3             7
     6     2             4
    12     1            13
- hence a(12) = min({4, 7, 13}) = 4.

Maple

a:= n-> min(seq(Bits[Xor](d, n/d), d=numtheory[divisors](n))):
seq(a(n), n=1..100);  # Alois P. Heinz, Oct 09 2019

Mathematica

mvx[n_]:=Min[BitXor[#, n/#]&/@Divisors[n]]; Array[mvx, 80] (* Harvey P. Dale, Nov 04 2019 *)

Prog

(PARI) a(n) = vecmin(apply(d -> bitxor(d, n/d), divisors(n)))

Crossrefs

See A328176 and A328177 for similar sequences.

Cf. A178910.

Sequence in context: A224825 A341413 A290794 * A127571 A194662 A143612

Adjacent sequences: A328175 A328176 A328177 * A328179 A328180 A328181

Keyword

nonn,base

Author

Rémy Sigrist, Oct 06 2019

Status

approved