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 A323830 #32 Jul 09 2025 04:48:30 %S A323830 1,2,4,8,16,32,64,128,256,512,1024,2048,4096,8192,16384,32768,636, %T A323830 1272,25,50,1,2,4,8,16,32,64,128,256,512,1024,2048,4096,8192,16384, %U A323830 32768,636,1272,25,50,1,2,4,8,16,32,64,128,256,512,1024,2048,4096,8192,16384,32768,636,1272,25,50 %N A323830 a(0) = 1; thereafter a(n) is obtained by doubling a(n-1) and repeatedly deleting any string of identical digits. %C A323830 Periodic with period length 20. %C A323830 Conjecture: If we start with any nonnegative number, and repeatedly double it and apply the "repeatedly delete any run of identical digits" operation described here, we eventually reach one of 0, 1, or 5. %C A323830 In other words, the conjecture is that eventually we reach 0 or join the trajectory shown here or the trajectory shown in A323831. %C A323830 The number of steps to reach 0, 1, or 5 is given in A323832. %H A323830 Colin Barker, <a href="/A323830/b323830.txt">Table of n, a(n) for n = 0..1000</a> %H A323830 <a href="/index/Rec#order_20">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1). %F A323830 From _Colin Barker_, Feb 03 2019: (Start) %F A323830 G.f.: (1 + 2*x)*(1 + 4*x^2 + 16*x^4 + 64*x^6 + 256*x^8 + 1024*x^10 + 4096*x^12 + 16384*x^14 + 636*x^16 + 25*x^18) / (1 - x^20). %F A323830 a(n) = a(n-20) for n>19. %F A323830 (End) %F A323830 a(n+1) = A321801(2*a(n)). For general numbers, the "repeatedly delete any run of identical digits" operation corresponds to repeatedly applying A321801. - _Chai Wah Wu_, Feb 11 2019 %e A323830 After a(15) = 32768 we get 65536 which becomes 636 after deleting "55". Then doubling 636 we get 1272, then 2544 which becomes 25 after deleting "44", then 50, then 100 which becomes 1 after deleting "00", and now the sequence repeats. %t A323830 dad[n_]:=FromDigits[FixedPoint[Flatten[Select[Split[#],Length[#]==1&]]&,IntegerDigits[2n]]];NestList[dad,1,100] (* _Paolo Xausa_, Nov 14 2023 *) %o A323830 (PARI) Vec((1 + 2*x)*(1 + 4*x^2 + 16*x^4 + 64*x^6 + 256*x^8 + 1024*x^10 + 4096*x^12 + 16384*x^14 + 636*x^16 + 25*x^18) / (1 - x^20) + O(x^40)) \\ _Colin Barker_, Feb 03 2019 %Y A323830 Cf. A000079, A320487, A321801, A321802, A323831, A323832. %Y A323830 See A035615 for a classic related base-2 sequence. %K A323830 nonn,base,easy %O A323830 0,2 %A A323830 _N. J. A. Sloane_, Feb 03 2019, following a suggestion from _Yukun Yao_