A245530 -id:A245530 - cp's OEIS frontend

A245499 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.

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, 22, 12, 15, 20, 13, 18, 26, 14, 15, 18, 20, 21, 15, 18, 20, 24, 16, 17, 18, 34, 18, 24, 27, 19, 32, 38, 20, 24, 30, 21, 27, 28, 22, 24, 33, 23, 32, 46, 24, 27, 32

Offset: 1

Reinhard Zumkeller, Jul 25 2014

A066400(n) = length of n-th row.
A006255(n) = T(n,A066400(n)), last term in n-th row.
A245530(n) = A066401(n)^2 = product of n-th row.
For n > 2: A245508(n) = T(A000040(n),2).
T(n,k) denote b_k in the definition given in A006255.
From David A. Corneth, Oct 22 2016 and Oct 25 2016: (Start)
Frequency of n in this sequence: 1, 1, 2, 1, 1, 3, 1, 5, 1, 3, 1, 4, 1, 2, ... See A277606.
Primes and squares occur once in this sequence except for 3 which occurs twice.
In the first 10000 rows, 9522 occurs most often and appears 60 times. 6498 is a close second with 59 occurrences.
(End)

			.    n |  Row(n)                | A066400(n) | A245530(n) | A066401(n)
. -----+------------------------+------------+------------+-----------
.    1 |  [1]                   |          1 |          1 |          1
.    2 |  [2, 3, 6]             |          3 |         36 |          6
.    3 |  [3, 6, 8]             |          3 |        144 |         12
.    4 |  [4]                   |          1 |          4 |          2
.    5 |  [5, 8, 10]            |          3 |        400 |         20
.    6 |  [6, 8, 12]            |          3 |        576 |         24
.    7 |  [7, 8, 14]            |          3 |        784 |         28
.    8 |  [8, 10, 12, 15]       |          4 |      14400 |        120
.    9 |  [9]                   |          1 |          9 |          3
.   10 |  [10, 12, 15, 18]      |          4 |      32400 |        180
.   11 |  [11, 18, 22]          |          3 |       4356 |         66
.   12 |  [12, 15, 20]          |          3 |       3600 |         60
.   13 |  [13, 18, 26]          |          3 |       6084 |         78
.   14 |  [14, 15, 18, 20, 21]  |          5 |    1587600 |       1260
.   15 |  [15, 18, 20, 24]      |          4 |     129600 |        360
.   16 |  [16]                  |          1 |         16 |          4
.   17 |  [17, 18, 34]          |          3 |      10404 |        102
.   18 |  [18, 24, 27]          |          3 |      11664 |        108
.   19 |  [19, 32, 38]          |          3 |      23104 |        152
.   20 |  [20, 24, 30]          |          3 |      14400 |        120
.   21 |  [21, 27, 28]          |          3 |      15876 |        126
.   22 |  [22, 24, 33]          |          3 |      17424 |        132
.   23 |  [23, 32, 46]          |          3 |      33856 |        184
.   24 |  [24, 27, 32]          |          3 |      20736 |        144
.   25 |  [25]                  |          1 |         25 |          5 .

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

Peter Kagey, Table of n, a(n) for n = 1..44055 (Derived from David A. Corneth's unflattened file)
David A. Corneth, First 10000 rows unflattened

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

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 *)

Following a suggestion of Peter Kagey, definition clarified by Reinhard Zumkeller, Nov 28 2014. Also removed erroneous program and b-file.

A066401 Square root of b_1b_2...*b_t corresponding to smallest values of t in R. L. Graham's sequence (A006255).

Original entry on oeis.org

1, 6, 12, 2, 20, 24, 28, 120, 3, 180, 66, 60, 78, 1260, 360, 4, 102, 108, 152, 120, 126, 132, 184, 144, 5, 936, 5040, 1120, 232, 210, 248, 240, 9240, 2040, 1680, 6, 370, 342, 312, 300, 410, 336, 430, 330, 360, 414, 470, 360, 7, 420, 25704, 196560, 636, 3780

Offset: 1

N. J. A. Sloane, Dec 25 2001

a(n) = A000196(A245530(n)). - Reinhard Zumkeller, Jul 25 2014

			a(2) = 6 because the best such sequence is 2,3,6 for which the product is 36 = 6^2.

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 147.

Peter Kagey, Table of n, a(n) for n = 1..1000 (First 125 terms from Reinhard Zumkeller)
R. L. Graham, Bijection between integers and composites, Problem 1242, Math. Mag., 60 (1987), p. 180.

Cf. A000196, A006255, A066400, A245499, A245530.

a066401 = a000196 . a245530  -- Reinhard Zumkeller, Jul 25 2014

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]]; Sqrt[Times @@ k], {n, 22}] (* Michael De Vlieger, Oct 26 2016 *)

More terms from Larry Reeves (larryr(AT)acm.org), Jan 06 2005

More terms from Joshua Zucker, May 18 2006

A302490 Fewest number of distinct prime factors in any product of a_1a_2...*a_t where n = a_1 < a_2 < ... < a_t = A006255(n) and the product is square.

Original entry on oeis.org

0, 2, 2, 1, 2, 2, 2, 3, 1, 3, 3, 3, 3, 4, 3, 1, 2, 2, 2, 3, 3, 3, 2, 2, 1, 3, 4, 3, 2, 4, 2, 3, 5, 4, 4, 2, 3, 3, 3, 3, 3, 3, 3, 4, 3, 3, 3, 3, 1, 4, 4, 5, 3, 4, 5, 3, 4

Offset: 1

Peter Kagey, Apr 08 2018

a(n^2) = A001221(n).

			a(14) = 4 because 14 * 15 * 16 * 18 * 20 * 21 has four distinct prime factors (2, 3, 5, and 7) and no other square product of a strictly increasing sequence starting at 14 and ending at 21 has fewer distinct prime factors.

Cf. A001221, A006255, A280244.

Cf. A066400, A245530.

cp's OEIS Frontend

A245499 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.

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A066401 Square root of b_1b_2...*b_t corresponding to smallest values of t in R. L. Graham's sequence (A006255).

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A302490 Fewest number of distinct prime factors in any product of a_1a_2...*a_t where n = a_1 < a_2 < ... < a_t = A006255(n) and the product is square.

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cp's OEIS Frontend

A245499 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.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

Extensions

A066401 Square root of b_1*b_2*...*b_t corresponding to smallest values of t in R. L. Graham's sequence (A006255).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Mathematica

Extensions

A302490 Fewest number of distinct prime factors in any product of a_1*a_2*...*a_t where n = a_1 < a_2 < ... < a_t = A006255(n) and the product is square.

Views

Author

Keywords

Comments

Examples

Crossrefs

A066401 Square root of b_1b_2...*b_t corresponding to smallest values of t in R. L. Graham's sequence (A006255).

A302490 Fewest number of distinct prime factors in any product of a_1a_2...*a_t where n = a_1 < a_2 < ... < a_t = A006255(n) and the product is square.