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a(1) = 2; for n > 1, a(n) is the smallest positive number that has not yet appeared such that the number of 1-bits in the binary expansion of a(n) equals the number of proper divisors of a(n-1).
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%I #12 Jul 04 2022 20:49:20

%S 2,4,3,8,7,16,15,11,32,31,64,63,47,128,127,256,255,191,512,511,13,

%T 1024,1023,223,2048,2047,14,19,4096,4095,8388607,21,22,25,5,8192,8191,

%U 16384,16383,239,32768,32767,247,26,28,55,35,37,65536,65535,49151,38,41,131072,131071,262144,262143

%N a(1) = 2; for n > 1, a(n) is the smallest positive number that has not yet appeared such that the number of 1-bits in the binary expansion of a(n) equals the number of proper divisors of a(n-1).

%C This sequence is similar to A355374 but the rules for determining a(n) are reversed. The only fixed point in the first 145 terms is a(3) = 3. It is unknown if all numbers eventually appear. The last known term is a(145) which is a 154 digit number whose complete factorization is unknown.

%H Scott R. Shannon, <a href="/A355482/b355482.txt">Table of n, a(n) for n = 1..145</a>

%e a(7) = 15 = 1111_2 as a(6) = 16 which has four proper divisors, and 15 is the smallest unused number that has four 1-bits in its binary expansion.

%Y Cf. A355483 (all divisors), A355374, A000120, A032741, A005179, A027751.

%K nonn,base

%O 1,1

%A _Scott R. Shannon_, Jul 03 2022