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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).

%I A306352 #14 Feb 11 2019 00:35:43
%S A306352 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,
%T A306352 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,
%U A306352 23,18,9,2,31,4,5,14,63,76,127,2,115,6,15,2,127,8,13
%N 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).
%C A306352 This sequence has similarities with A167234.
%C A306352 Will every nonnegative integer appear in the sequence?
%H A306352 Rémy Sigrist, <a href="/A306352/b306352.txt">Table of n, a(n) for n = 1..10000</a>
%H A306352 Rémy Sigrist, <a href="/A306352/a306352.png">Colored logarithmic scatterplot of the sequence for n = 1..2^19</a> (where the color is function of floor(n / 2^A070939(a(n)))).
%F A306352 a(2^k) = 2^k - 1 for any k >= 0.
%F A306352 a(n) = 2 iff n belongs to A002145.
%F A306352 a(n) <= A218388(n).
%F A306352 a(n) AND A218388(n) = a(n).
%F A306352 A000120(a(n)) = 1 iff n is a prime number.
%F A306352 Apparently:
%F A306352 - a(3^k) belongs to A131130 for any k > 0,
%F A306352 - a(5^k) belongs to A028399 for any k >= 0.
%e A306352 For n = 15:
%e A306352 - the divisors of 15 are: 1, 3, 5 and 15,
%e A306352 - their values under the mapping d -> d AND k for k = 0..6 are:
%e A306352   k\d|  1  3  5  15
%e A306352   ---+-------------
%e A306352     0|  0  0  0  0
%e A306352     1|  1  1  1  1
%e A306352     2|  0  2  0  2
%e A306352     3|  1  3  1  3
%e A306352     4|  0  0  4  4
%e A306352     5|  1  1  5  5
%e A306352     6|  0  2  4  6
%e A306352 - the first row with 4 distinct values corresponds to k = 6,
%e A306352 - hence a(15) = 6.
%o A306352 (PARI) a(n) = my (d=divisors(n)); for (m=0, oo, if (#Set(apply(v -> bitand(v, m), d))==#d, return (m)))
%Y A306352 Cf. A000120, A002145, A028399, A070939, A131130, A167234, A218388.
%K A306352 nonn,base
%O A306352 1,3
%A A306352 _Rémy Sigrist_, Feb 09 2019