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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). 1
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified December 5 13:26 EST 2019. Contains 329751 sequences. (Running on oeis4.)