A245499 - OEIS

%I #35 May 02 2017 22:17:17

%S 1,2,3,6,3,6,8,4,5,8,10,6,8,12,7,8,14,8,10,12,15,9,10,12,15,18,11,18,

%T 22,12,15,20,13,18,26,14,15,18,20,21,15,18,20,24,16,17,18,34,18,24,27,

%U 19,32,38,20,24,30,21,27,28,22,24,33,23,32,46,24,27,32

%N Table read by rows: n-th row contains the factors which occur when constructing R. L. Graham's sequence A006255, such that the number of factors and also the product is minimal.

%C A066400(n) = length of n-th row.

%C A006255(n) = T(n,A066400(n)), last term in n-th row.

%C A245530(n) = A066401(n)^2 = product of n-th row.

%C For n > 2: A245508(n) = T(A000040(n),2).

%C T(n,k) denote b_k in the definition given in A006255.

%C From _David A. Corneth_, Oct 22 2016 and Oct 25 2016: (Start)

%C Frequency of n in this sequence: 1, 1, 2, 1, 1, 3, 1, 5, 1, 3, 1, 4, 1, 2, ... See A277606.

%C Primes and squares occur once in this sequence except for 3 which occurs twice.

%C In the first 10000 rows, 9522 occurs most often and appears 60 times. 6498 is a close second with 59 occurrences.

%C (End)

%D R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 2nd. ed., Problem 4.39, pages 147, 616, 533.

%H Peter Kagey, <a href="/A245499/b245499.txt">Table of n, a(n) for n = 1..44055</a> (Derived from David A. Corneth's unflattened file)

%H David A. Corneth, <a href="/A245499/a245499_1.gp.txt">First 10000 rows unflattened</a>

%e .    n |  Row(n)                | A066400(n) | A245530(n) | A066401(n)

%e . -----+------------------------+------------+------------+-----------

%e .    1 |  [1]                   |          1 |          1 |          1

%e .    2 |  [2, 3, 6]             |          3 |         36 |          6

%e .    3 |  [3, 6, 8]             |          3 |        144 |         12

%e .    4 |  [4]                   |          1 |          4 |          2

%e .    5 |  [5, 8, 10]            |          3 |        400 |         20

%e .    6 |  [6, 8, 12]            |          3 |        576 |         24

%e .    7 |  [7, 8, 14]            |          3 |        784 |         28

%e .    8 |  [8, 10, 12, 15]       |          4 |      14400 |        120

%e .    9 |  [9]                   |          1 |          9 |          3

%e .   10 |  [10, 12, 15, 18]      |          4 |      32400 |        180

%e .   11 |  [11, 18, 22]          |          3 |       4356 |         66

%e .   12 |  [12, 15, 20]          |          3 |       3600 |         60

%e .   13 |  [13, 18, 26]          |          3 |       6084 |         78

%e .   14 |  [14, 15, 18, 20, 21]  |          5 |    1587600 |       1260

%e .   15 |  [15, 18, 20, 24]      |          4 |     129600 |        360

%e .   16 |  [16]                  |          1 |         16 |          4

%e .   17 |  [17, 18, 34]          |          3 |      10404 |        102

%e .   18 |  [18, 24, 27]          |          3 |      11664 |        108

%e .   19 |  [19, 32, 38]          |          3 |      23104 |        152

%e .   20 |  [20, 24, 30]          |          3 |      14400 |        120

%e .   21 |  [21, 27, 28]          |          3 |      15876 |        126

%e .   22 |  [22, 24, 33]          |          3 |      17424 |        132

%e .   23 |  [23, 32, 46]          |          3 |      33856 |        184

%e .   24 |  [24, 27, 32]          |          3 |      20736 |        144

%e .   25 |  [25]                  |          1 |         25 |          5 .

%t Table[k = 0; While[Length@ # == 0 &@ Set[f, Select[Rest@ Subsets@ Range@ k, IntegerQ@ Sqrt[n (Times @@ # &[n + #])] &]], k++]; If[IntegerQ@ Sqrt@ n, k = {n}, k = n + Prepend[First@ f, 0]]; k, {n, 22}] (* _Michael De Vlieger_, Oct 26 2016 *)

%Y Cf. A006255, A006530, A007913, A010051, A010052, A020639, A049084, A066400 (row lengths), A066401, A089229, A151800, A245508, A245530 (row products), A277606.

%K nonn,tabf

%O 1,2

%A _Reinhard Zumkeller_, Jul 25 2014

%E Following a suggestion of _Peter Kagey_, definition clarified by _Reinhard Zumkeller_, Nov 28 2014. Also removed erroneous program and b-file.