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A208884 a(n) = (a(n-1) + n)/2^k where 2^k is the largest power of 2 dividing a(n-1) + n, for n>1 with a(1)=1.

%I A208884 #41 Jun 28 2020 02:26:11
%S A208884 1,3,3,7,3,9,1,9,9,19,15,27,5,19,17,33,25,43,31,51,9,31,27,51,19,45,9,
%T A208884 37,33,63,47,79,7,41,19,55,23,61,25,65,53,95,69,113,79,125,43,91,35,
%U A208884 85,17,69,61,115,85,141,99,157,27,87,37,99,81,145,105,171
%N A208884 a(n) = (a(n-1) + n)/2^k where 2^k is the largest power of 2 dividing a(n-1) + n, for n>1 with a(1)=1.
%C A208884 In other words, to get a(n), add n to a(n-1) and compute the odd part (A000265) of the sum. - _Ralf Stephan_, Oct 27 2013
%C A208884 POSITIONS of odd numbers in the initial 7000000 terms begin:
%C A208884 1: [1, 7, 69, 285, 3601, 5167, 92989, 112651, 6933175, ...];
%C A208884 3: [2, 3, 5, 613, 8461, 46749, 81237, 102171, 126661, 3309589, ...];
%C A208884 5: [13, 97, 2431, 92095, ...];
%C A208884 7: [4, 33, 3167, 78095, 2723179, ...];
%C A208884 9: [6, 8, 9, 21, 27, 303, 2017, 3239, 3765, 6753, 28387, 251451, ...];
%C A208884 11: [75, 15823, 28221, 4091959, 5820487, ...];
%C A208884 13: [22975, 42391, 3729249, ...];
%C A208884 15: [11, 22587, 2527579, 6954893, ...];
%C A208884 17: [15, 51, 3121, 13433, 74763, 376853, 576439, 896899, ...];
%C A208884 19: [10, 14, 25, 35, 291, 77747, 757319, 1227595, 2307099, ...];
%C A208884 21: [1417, 1557, 712229, 2563807, ...];
%C A208884 23: [37, 127, 609, 2211, 5563, 199901, ...];
%C A208884 25: [17, 39, 221, 1145, 3425, 17593, 4318897, ...];
%C A208884 27: [12, 23, 59, 73, 289, 1149, 3393, 20439, 37107, ...];
%C A208884 29: [573, 33315, 61505, 467047, 491359, 1170709, 1492309, 2498593, 3017011, ...];
%C A208884 31: [19, 22, 229, 409, 6199, 60529, 3602675, 4108215, 4604929, ...]; ...
%C A208884 From _Ya-Ping Lu_, Jun 25 2020: (Start)
%C A208884 Conjecture: For any given odd number m, there exists a number n_max such that all odd numbers <= m can be found in the sequence a(n) with n <= n_max. For example:
%C A208884 m = 1, n_max = 1;
%C A208884 m = 3, n_max = 2;
%C A208884 m = 5, n_max = 13;
%C A208884 m = 11, n_max = 75
%C A208884 m = 13, n_max = 22975;
%C A208884 m = 305, n_max = 1025715;
%C A208884 m = 749, n_max = 14695985;
%C A208884 m = 795, n_max = 150788015;
%C A208884 m = 7525, n_max = 31129547917;
%C A208884 ...
%C A208884 If the conjecture above is true, this sequence contains all odd numbers. (End)
%H A208884 Paul D. Hanna, <a href="/A208884/b208884.txt">Table of n, a(n) for n = 1..10000</a>
%H A208884 Rémy Sigrist, <a href="/A208884/a208884.png">Colored scatterplot of the first 100000 terms</a> (where the color is function of the parity of n)
%e A208884 a(2) = 1 + 2 = 3;
%e A208884 a(3) = (3 + 3)/2 = 3;
%e A208884 a(4) = 3 + 4 = 7;
%e A208884 a(5) = (7 + 5)/4 = 3;
%e A208884 a(6) = 3 + 6 = 9;
%e A208884 a(7) = (9 + 7)/16 = 1; ...
%t A208884 a[1]=1; a[n_] := a[n] = #/2^IntegerExponent[#, 2] &@ (n + a[n-1]); Array[a, 70] (* _Giovanni Resta_, Jun 25 2020 *)
%o A208884 (PARI) {a(n)=if(n==1, 1, (a(n-1)+n)/2^valuation(a(n-1)+n,2))}
%o A208884 (PARI) {A=vector(1024); a(n)=A[n]=if(n==1, 1, (A[n-1]+n)/2^valuation(A[n-1]+n,2))}
%o A208884 for(n=1,#A,print1(a(n),", "))
%Y A208884 Cf. A069834, A090895, A114216, A335817.
%K A208884 nonn
%O A208884 1,2
%A A208884 _Paul D. Hanna_, Mar 02 2012