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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.
3
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, 37, 33, 63, 47, 79, 7, 41, 19, 55, 23, 61, 25, 65, 53, 95, 69, 113, 79, 125, 43, 91, 35, 85, 17, 69, 61, 115, 85, 141, 99, 157, 27, 87, 37, 99, 81, 145, 105, 171
OFFSET
1,2
COMMENTS
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
POSITIONS of odd numbers in the initial 7000000 terms begin:
1: [1, 7, 69, 285, 3601, 5167, 92989, 112651, 6933175, ...];
3: [2, 3, 5, 613, 8461, 46749, 81237, 102171, 126661, 3309589, ...];
5: [13, 97, 2431, 92095, ...];
7: [4, 33, 3167, 78095, 2723179, ...];
9: [6, 8, 9, 21, 27, 303, 2017, 3239, 3765, 6753, 28387, 251451, ...];
11: [75, 15823, 28221, 4091959, 5820487, ...];
13: [22975, 42391, 3729249, ...];
15: [11, 22587, 2527579, 6954893, ...];
17: [15, 51, 3121, 13433, 74763, 376853, 576439, 896899, ...];
19: [10, 14, 25, 35, 291, 77747, 757319, 1227595, 2307099, ...];
21: [1417, 1557, 712229, 2563807, ...];
23: [37, 127, 609, 2211, 5563, 199901, ...];
25: [17, 39, 221, 1145, 3425, 17593, 4318897, ...];
27: [12, 23, 59, 73, 289, 1149, 3393, 20439, 37107, ...];
29: [573, 33315, 61505, 467047, 491359, 1170709, 1492309, 2498593, 3017011, ...];
31: [19, 22, 229, 409, 6199, 60529, 3602675, 4108215, 4604929, ...]; ...
From Ya-Ping Lu, Jun 25 2020: (Start)
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:
m = 1, n_max = 1;
m = 3, n_max = 2;
m = 5, n_max = 13;
m = 11, n_max = 75
m = 13, n_max = 22975;
m = 305, n_max = 1025715;
m = 749, n_max = 14695985;
m = 795, n_max = 150788015;
m = 7525, n_max = 31129547917;
...
If the conjecture above is true, this sequence contains all odd numbers. (End)
LINKS
Rémy Sigrist, Colored scatterplot of the first 100000 terms (where the color is function of the parity of n)
EXAMPLE
a(2) = 1 + 2 = 3;
a(3) = (3 + 3)/2 = 3;
a(4) = 3 + 4 = 7;
a(5) = (7 + 5)/4 = 3;
a(6) = 3 + 6 = 9;
a(7) = (9 + 7)/16 = 1; ...
MATHEMATICA
a[1]=1; a[n_] := a[n] = #/2^IntegerExponent[#, 2] &@ (n + a[n-1]); Array[a, 70] (* Giovanni Resta, Jun 25 2020 *)
PROG
(PARI) {a(n)=if(n==1, 1, (a(n-1)+n)/2^valuation(a(n-1)+n, 2))}
(PARI) {A=vector(1024); a(n)=A[n]=if(n==1, 1, (A[n-1]+n)/2^valuation(A[n-1]+n, 2))}
for(n=1, #A, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Mar 02 2012
STATUS
approved