A339239 - OEIS

%I #6 Nov 28 2020 09:11:23

%S 1,6,63,250,131070,1048574,33554426,17179869180

%N Binary self numbers (A010061) with a record gap to the next binary self number.

%C The corresponding gaps are 3, 7, 8, 20, 21, 24, 37, 42, ...

%C a(9) <= 288230376151711738.

%C Apparently, the records gaps occur for pairs of consecutive binary self numbers with a power of 2 between them. If this is generally true, then the next terms are 288230376151711738, 147573952589676412923, 37778931862957161709564, 10633823966279326983230456482242756602, 5444517870735015415413993718908291383294, 43556142965880123323311949751266331066367, ..., with the corresponding gaps 70, 77, 83, 135, 136, 137, ...

%e The first 4 binary self numbers are 1, 4, 6 and 13. The gaps between them are 3, 2 and 7. The record gaps are 3 and 7, and the corresponding terms are 1 and 6.

%t s[n_] := n + DigitCount[n, 2, 1]; selfQ[n_] := AllTrue[Range[n, n - Floor@Log2[n], -1], s[#] != n &]; dm = 0; seq = {}; n1 = 1; Do[If[selfQ[n], d = n - n1; If[d > dm, dm = d; AppendTo[seq, n1]]; n1 = n], {n, 2, 150000}]; seq

%Y Cf. A010061, A339216.

%K nonn,base,more

%O 1,2

%A _Amiram Eldar_, Nov 28 2020