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URL: https://oeis.org/A306352

⇱ A306352 - OEIS

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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
OFFSET
1,3
COMMENTS
This sequence has similarities with A167234.
Will every nonnegative integer appear in the sequence?
LINKS
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)))
KEYWORD
nonn,base
AUTHOR
Rémy Sigrist, Feb 09 2019
STATUS
approved