This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A387423 #16 Sep 02 2025 01:41:55
%S A387423 0,1,1,2,2,0,2,3,3,2,3,2,2,2,2,4,2,1,3,1,3,3,4,3,3,4,4,0,2,4,4,5,5,5,
%T A387423 5,4,2,5,5,3,2,5,3,3,4,4,5,4,4,5,5,3,2,4,5,3,5,5,3,5,2,4,4,6,5,4,3,6,
%U A387423 6,4,4,6,2,6,6,4,6,5,5,4,6,6,3,6,6,5,6,2,2,6,6,4,5,5,6,5,2,6,6,4,2,4,4,1,4
%N A387423 The length of binary expansion of n minus the length of the maximal common prefix of the binary expansions of n and sigma(n), where sigma is the sum of divisors function.
%C A387423 Positions of 0's in this sequence is given by such numbers n that sigma(n) = 2^k * n + r, for some n >= 1, k >= 0, 0 <= r < 2^k. These would include also quasi-perfect numbers and their generalizations, numbers n such that sigma(n) = 2^k * n + 2^k - 1, for some n > 1, k > 0 (see comments in A332223), if such numbers exist. However, it is conjectured that there are no other zeros than those given by A336702.
%H A387423 Antti Karttunen, <a href="/A387423/b387423.txt">Table of n, a(n) for n = 1..65537</a>
%H A387423 <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>.
%H A387423 <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>.
%F A387423 a(n) = (1+A000523(n)) - A387422(n).
%o A387423 (PARI) A387423(n) = { my(a=binary(n), b=binary(sigma(n)), i=1); while(i<=#a,if(a[i]!=b[i],return(#a-(i-1))); i++); (0); };
%Y A387423 Cf. A000203, A000523, A332223, A336700, A336701, A336702 (conjectured positions of 0's), A387422.
%Y A387423 Cf. also A347381, A387413.
%K A387423 nonn,base,new
%O A387423 1,4
%A A387423 _Antti Karttunen_, Sep 01 2025