A306352 a(n) is the least k >= 0 such that all the positive divisors of n have a distinct value under the mapping d -> d AND k (where AND denotes the bitwise AND operator).

0, 1, 2, 3, 4, 7, 2, 7, 10, 13, 2, 15, 4, 5, 6, 15, 16, 31, 2, 29, 6, 7, 2, 31, 12, 9, 10, 11, 4, 15, 2, 31, 42, 49, 6, 63, 4, 7, 6, 63, 8, 15, 2, 14, 14, 5, 2, 63, 18, 29, 18, 21, 4, 31, 6, 23, 18, 9, 2, 31, 4, 5, 14, 63, 76, 127, 2, 115, 6, 15, 2, 127, 8, 13

Offset

1,3

Comments

This sequence has similarities with A167234.

Will every nonnegative integer appear in the sequence?

Links

Rémy Sigrist, Table of n, a(n) for n = 1..10000

Rémy Sigrist, Colored logarithmic scatterplot of the sequence for n = 1..2^19 (where the color is function of floor(n / 2^A070939(a(n)))).

Formula

a(2^k) = 2^k - 1 for any k >= 0.

a(n) = 2 iff n belongs to A002145.

a(n) <= A218388(n).

a(n) AND A218388(n) = a(n).

A000120(a(n)) = 1 iff n is a prime number.

Apparently:

- a(3^k) belongs to A131130 for any k > 0,

- a(5^k) belongs to A028399 for any k >= 0.

Example

For n = 15:
- the divisors of 15 are: 1, 3, 5 and 15,
- their values under the mapping d -> d AND k for k = 0..6 are:
  k\d|  1  3  5  15
  ---+-------------
    0|  0  0  0  0
    1|  1  1  1  1
    2|  0  2  0  2
    3|  1  3  1  3
    4|  0  0  4  4
    5|  1  1  5  5
    6|  0  2  4  6
- the first row with 4 distinct values corresponds to k = 6,
- hence a(15) = 6.

Prog

(PARI) a(n) = my (d=divisors(n)); for (m=0, oo, if (#Set(apply(v -> bitand(v, m), d))==#d, return (m)))

Crossrefs

Cf. A000120, A002145, A028399, A070939, A131130, A167234, A218388.

Sequence in context: A021903 A274767 A058315 * A072717 A139072 A021430

Adjacent sequences: A306349 A306350 A306351 * A306353 A306354 A306355

Keyword

nonn,base

Author

Rémy Sigrist, Feb 09 2019

Status

approved