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 A281978 Lexicographically earliest sequence of distinct terms such that, for any n>0, a(2n) is divisible by a(2n-1) and by a(2n+1). 9
 1, 4, 2, 6, 3, 15, 5, 20, 10, 40, 8, 24, 12, 36, 9, 54, 18, 90, 30, 120, 60, 180, 45, 135, 27, 162, 81, 324, 108, 216, 72, 144, 16, 64, 32, 96, 48, 240, 80, 320, 160, 640, 128, 384, 192, 576, 288, 864, 432, 1296, 648, 1944, 243, 972, 486, 1458, 729, 3645, 405 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS To compute a(2n) and a(2n+1): we take the least unseen multiple of a(2n-1) with an unseen proper divisor: the multiple gives a(2n) and the least proger divisor gives a(2n+1). The first multiple of 2 occurs at n=2: a(2)=4, and a(3)=2. The first multiple of 3 occurs at n=4: a(4)=6, and a(5)=3, The first multiple of 5 occurs at n=6: a(6)=15, and a(7)=5. The first multiple of 7 occurs at n=454: a(454)=5511240, and a(455)=7. The first multiple of 11 occurs at n=889838: a(889838)=627667978163491186346557440000000000000, and a(889839)=11. For n>1, let b(n)=least k>0 such that a(n+k)<>a(n)*a(k+1); the first records for b are: n       b(n)     a(n) ------  -------  ---- 2             1  2^2 7             3  5 19            4  2*3*5 33           14  2^4 73           27  5^2 455         243  7 1439        248  7^2 3069        275  7^3 10567       276  7^5 41709       768  7^8 85179      1169  7^10 889839  >110162  11 Conjectures: - All prime numbers appear in this sequence, in increasing order, - The derived sequence b is unbounded, - This sequence is a permutation of the natural numbers. LINKS Rémy Sigrist, Table of n, a(n) for n = 1..25000 Rémy Sigrist, PARI program for A281978 Rémy Sigrist, Logarithmic scatterplot of the first million terms EXAMPLE The first terms, alongside their p-adic valuations with respect to p=2, 3, 5 and 7 (with 0's omitted), are: n       a(n)  v2  v3  v5  v7 ---  -------  --  --  --  -- 1          1 2          4   2 3          2   1 4          6   1   1 5          3       1 6         15       1   1 7          5           1 8         20   2       1 9         10   1       1 10        40   3       1 11         8   3 12        24   3   1 13        12   2   1 14        36   2   2 15         9       2 16        54   1   3 17        18   1   2 18        90   1   2   1 19        30   1   1   1 20       120   3   1   1 21        60   2   1   1 22       180   2   2   1 23        45       2   1 24       135       3   1 ... 451   524880   4   8   1 452  1574640   4   9   1 453   787320   3   9   1 454  5511240   3   9   1   1 455        7               1 456       28   2           1 457       14   1           1 458       42   1   1       1 CROSSREFS Cf. A036552 (a(2n) is divisible by a(2n-1)). Sequence in context: A169839 A308712 A275845 * A325887 A209877 A273667 Adjacent sequences:  A281975 A281976 A281977 * A281979 A281980 A281981 KEYWORD nonn AUTHOR Rémy Sigrist, Feb 04 2017 STATUS approved

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Last modified June 6 17:33 EDT 2020. Contains 334831 sequences. (Running on oeis4.)