A295368 - OEIS

%I #23 Feb 19 2018 14:46:35

%S 1,2,3,4,3,10,3,8,7,10,3,40,3,10,15,16,3,42,3,36,15,10,3,160,7,10,15,

%T 36,3,178,3,32,15,10,15,328,3,10,15,144,3,170,3,36,63,10,3,640,7,42,

%U 15,36,3,170,15,144,15,10,3,2696,3,10,63,64,15,170,3,36,15

%N For any number n > 0 with s divisors, say d_1, d_2, ..., d_s such that d_1 = 1 < d_2 < ... < d_s = n, the binary representation of a(n) is (d_1 mod 2, d_2 mod 2, ..., d_s mod 2).

%C This sequence encodes in binary the parity of the divisors of a number.

%C For any n > 0, the binary representation of a(n) corresponds to the n-th row of A247795.

%C For any n > 0, n and a(n) have the same parity.

%H Rémy Sigrist, <a href="/A295368/b295368.txt">Table of n, a(n) for n = 1..10000</a>

%H Rémy Sigrist, <a href="/A295368/a295368.png">Colored logarithmic scatterplot of the first 100000 terms</a> (where the color is function of A000005(n))

%F a(2^k) = 2^k for any k >= 0.

%F a(p) = 3 iff p is an odd prime.

%F a(n) > 3 iff n is composite.

%F A070939(a(n)) = A000005(n) for any n > 0.

%F A000120(a(n)) = A001227(n) for any n > 0.

%F A023416(a(n)) = A183063(n) for any n > 0.

%F A000120(a(n)) = A023416(a(n)) iff n belongs to A016825.

%t Array[FromDigits[Mod[#, 2] & /@ Divisors@ #, 2] &, 69] (* _Michael De Vlieger_, Feb 18 2018 *)

%o (PARI) a(n) = fromdigits(apply(d -> d%2, divisors(n)), 2)

%Y Cf. A000005, A000120, A001227, A016825, A023416, A070939, A183063, A247795.

%K nonn,base

%O 1,2

%A _Rémy Sigrist_, Feb 18 2018