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

Author

Rémy Sigrist, Oct 06 2019

Keywords

• nonn
• base

Examples

			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.
		

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

Crossrefs

See A328176 and A328177 for similar sequences.

Cf. A178910.

Programs

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 *)

PARI

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

Formula

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

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