A006255 -id:A006255 - cp's OEIS frontend

A066400 Smallest values of t arising in R. L. Graham's sequence (A006255).

1, 3, 3, 1, 3, 3, 3, 4, 1, 4, 3, 3, 3, 5, 4, 1, 3, 3, 3, 3, 3, 3, 3, 3, 1, 4, 5, 4, 3, 3, 3, 3, 5, 4, 4, 1, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 3, 5, 6, 3, 4, 5, 3, 3, 4, 3, 5, 3, 4, 5, 1, 6, 5, 3, 3, 3, 5, 3, 5, 3, 3, 6, 3, 4, 5, 3, 3, 1, 3, 3, 4, 5, 3, 3, 3, 3, 6, 6, 5, 3, 3, 5, 3, 3, 6, 7, 1, 3, 6, 3, 5, 4

Offset: 1

N. J. A. Sloane, Dec 25 2001

nonn

Length of n-th row in table A245499. - Reinhard Zumkeller, Jul 25 2014
Indices of records are 1, 2, 8, 14, 52, 99, 589, 594, 595... (A277649) - Peter Kagey, Oct 24 2016
It is conjectured that 2 never appears in this sequence. a(n) = 2 if and only if A006255(n) = A072905(n). - Peter Kagey, Oct 25 2016
a(n) is three most of the time, then 5, then 6, then 4 for the first 1000 and the first 10000 terms. At n = 72, 78 and 85, a(n) is 4 or 5 and 4 and 5 occurred equally often so far. At 299, 301, 312, 322 and 403, a(n) is 4 or 6 and 4 and 6 occurred equally often so far. This doesn't happen for the first 10000 terms for 5 and 6. - David A. Corneth, Oct 25 2016

			a(2) = 3 because the best such sequence is 2,3,6 which has three terms.

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

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 1000 terms from Reinhard Zumkeller and Peter Kagey)
R. L. Graham, Bijection between integers and composites, Problem 1242, Math. Mag., 60 (1987), p. 180.

Cf. A006255, A066401, A072905.

Cf. A245499, A277649.

a066400 = length . a245499_row  -- Reinhard Zumkeller, Jul 25 2014

More terms from John W. Layman, Jul 14 2003

More terms from Joshua Zucker, May 18 2006

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.

Original entry on oeis.org

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.

A259527 a(n) gives the number of sequences n = b_1 < b_2 < ... < b_t = A006255(n) such that b_1b_2...*b_t is a perfect square.

Original entry on oeis.org

1, 2, 2, 1, 2, 2, 2, 2, 1, 2, 8, 2, 16, 2, 2, 1, 64, 2, 128, 4, 2, 4, 512, 2, 1, 4, 1, 2, 8192, 2, 8192, 4, 2, 16, 2, 1, 65536, 64, 4, 2, 524288, 8, 1048576, 4, 4, 128, 8388608, 2, 1, 1, 8, 2, 67108864, 4, 2, 2, 4, 256, 536870912, 2, 2147483648, 2048, 2, 1, 1

Offset: 1

Peter Kagey, Jun 29 2015

nonn

All terms are powers of 2.

			For a(20)=4 the solutions are:
s_0 = {20,24,30} with prod(s_0) = 120^2;
s_1 = {20,24,25,30} with prod(s_1) = 600^2;
s_2 = {20,21,24,27,28,30} with prod(s_2) = 15120^2;
s_3 = {20,21,24,25,27,28,30} with prod(s_3) = 75600^2.

Peter Kagey, Table of n, a(n) for n = 1..1000

Cf. A006255, A260510.

More terms from Alois P. Heinz, Jul 16 2015

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

A067565 Inverse of R. L. Graham's sequence (A006255), or zero if and only if n is a prime.

Original entry on oeis.org

1, 0, 0, 4, 0, 2, 0, 3, 9, 5, 0, 6, 0, 7, 8, 16, 0, 10, 0, 12, 14, 11, 0, 15, 25, 13, 18, 21, 0, 20, 0, 24, 22, 17, 27, 36, 0, 19, 26, 28, 0, 30, 0, 33, 32, 23, 0, 35, 49, 40, 34, 39, 0, 48, 44, 42, 38, 29, 0, 45, 0, 31, 50, 64, 52, 55, 0, 51, 46, 54

Offset: 1

Robert G. Wilson v, Jan 30 2002

nonn

a(n) = n if and only if n is a perfect square.

Also, the greatest m such that there exists a sequence m = a_1 < a_2 < ... < a_t = n such that a_1 * a_2 * ... * a_t is square. - Peter Kagey, Dec 27 2016

Peter Kagey, Table of n, a(n) for n = 1..5000

Cf. A006255.

A269045 Indices k such that A006255(k) != A070229(k); that is, the k-th term of R. L. Graham's sequence is not equal to k + lpf(k).

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 9, 10, 12, 15, 16, 18, 20, 24, 25, 27, 28, 30, 32, 35, 36, 40, 42, 45, 48, 49, 50, 54, 56, 60, 63, 64, 66, 70, 72, 75, 77, 80, 81, 84, 90, 91, 96, 98, 99, 100, 104, 105, 108, 110, 112, 120, 121, 125, 126, 128, 130, 132, 135, 140, 143, 144

Offset: 1

Peter Kagey, Feb 22 2016

nonn

Peter Kagey, Table of n, a(n) for n = 1..2500

Cf. A006255, A070229.

Complement of A255363.

A233421 Let m = n-th nonsquare = A000037(n); then a(n) = A006255(m).

Original entry on oeis.org

6, 8, 10, 12, 14, 15, 18, 22, 20, 26, 21, 24, 34, 27, 38, 30, 28, 33, 46, 32, 39, 35, 40, 58, 42, 62, 45, 44, 51, 48, 74, 57, 52, 50, 82, 56, 86, 55, 60, 69, 94, 54, 63, 68, 65, 106, 70, 66, 72, 76, 87, 118, 75, 122, 93, 77, 78, 80, 134, 85, 92, 84

Offset: 1

Peter Kagey, Dec 09 2013

nonn

			a(1) = A006255(A000037(1)) = A006255(2) = 6 because 2*3*6 = 6^2.
a(2) = A006255(A000037(2)) = A006255(3) = 8 because 3*6*8 = 12^2.

Peter Kagey, Table of n, a(n) for n = 1..5000
William Lowell Putnam Competition, Problem A2, 2013.
R. L. Graham, Bijection between integers and composites, Problem 1242, Math. Mag., 60 (1987), p. 180. [Note that unless you subscribe to JSTOR this link will only show page 178, which contains a different problem proposed by R. L. Graham. - _N. J. A. Sloane_, Jan 13 2014]

Arguments are numbers that are nonsquares: A000037.

This is A006255 with perfect squares omitted.

a(n) = A006255(A000037(n)). - Michel Marcus, Jan 07 2014

Edited by Michel Marcus and N. J. A. Sloane, Jan 13 2014

A255167 a(n) = A072905(n) - A006255(n).

Original entry on oeis.org

3, 2, 4, 5, 10, 12, 14, 3, 7, 22, 22, 7, 26, 35, 36, 9, 34, 5, 38, 15, 56, 55, 46, 22, 11, 65, 13, 23, 58, 78, 62, 5, 88, 85, 92, 13, 74, 95, 104, 40, 82, 112, 86, 44, 20, 115, 94, 21, 15, 9, 136, 52, 106, 26, 154, 54, 152, 145, 118, 60, 122, 155, 35, 17

Offset: 1

Peter Kagey, Feb 15 2015

nonn

a(n) is strictly nonnegative because A072905 is an upper bound on A006255.
A066400(n) = 2 if and only if a(n) = 0.
a(n) > 0 for all n <= 10000.

Peter Kagey, Table of n, a(n) for n = 1..10000

Cf. A072905, A006255, A066400.

a(n) = A072905(n) - A006255(n).

A255363 Numbers with the property that A006255(k) = A070229(k).

Original entry on oeis.org

5, 7, 11, 13, 14, 17, 19, 21, 22, 23, 26, 29, 31, 33, 34, 37, 38, 39, 41, 43, 44, 46, 47, 51, 52, 53, 55, 57, 58, 59, 61, 62, 65, 67, 68, 69, 71, 73, 74, 76, 78, 79, 82, 83, 85, 86, 87, 88, 89, 92, 93, 94, 95, 97, 101, 102, 103, 106, 107, 109, 111, 113, 114

Offset: 1

Peter Kagey, Feb 21 2015

nonn

A070229(n) is a lower bound of A006255(n) for all n in A102750.
This list contains all primes greater than 3 and no perfect squares.
Let k be a fixed integer, then k*p is found in this list for all sufficiently large primes p.

Peter Kagey, Table of n, a(n) for n = 1..5000

A280244 Lexicographically ordered list of sequences that meet the criteria for R. L. Graham's sequence: k = a_1 < a_2 < ... < a_t = A006255(k) and a_1a_2...*a_t is a square.

Original entry on oeis.org

1, 2, 3, 4, 6, 2, 3, 6, 3, 4, 6, 8, 3, 6, 8, 4, 5, 8, 9, 10, 5, 8, 10, 6, 8, 9, 12, 6, 8, 12, 7, 8, 9, 14, 7, 8, 14, 8, 9, 10, 12, 15, 8, 10, 12, 15, 9, 10, 12, 15, 16, 18, 10, 12, 15, 18, 11, 12, 14, 16, 21, 22, 11, 12, 14, 21, 22, 11, 12, 15, 16, 18, 20, 22

Offset: 1

Peter Kagey, Dec 29 2016

A259527(n) rows begin with n.

			[8,9,10,12,15] appears as a row in the table because A006255(8) = 15 and the product of the row is a square: 8*9*10*12*15 = 360^2.
Table begins:
  1;
  2,  3,  4,  6;
  2,  3,  6;
  3,  4,  6,  8;
  3,  6,  8;
  4;
  5,  8,  9, 10;
  5,  8, 10;
  6,  8,  9, 12;
  6,  8, 12;
  7,  8,  9, 14;
  7,  8, 14;
  8,  9, 10, 12, 15;
  8, 10, 12, 15;
  ...

Peter Kagey, Table of n, a(n) for n = 1..10000

Cf. A006255, A245499, A259527.

MapIndexed[With[{b = #1, a = First@ #2}, Reverse@ Select[Rest@ Subsets@ Range[a, b], And[SubsetQ[#, {a, b}], IntegerQ@ Sqrt[Times @@ #]] &]] &, #] &@ Table[k = 0; Which[IntegerQ@ Sqrt@ n, k, And[PrimeQ@ n, n > 3], k = n, True, While[Length@ Select[n Map[Times @@ # &, n + Rest@ Subsets@ Range@ k], IntegerQ@ Sqrt@# &] == 0, k++]]; k + n, {n, 16}] // Flatten (* Michael De Vlieger, Dec 30 2016 *)

cp's OEIS Frontend

A066400 Smallest values of t arising in R. L. Graham's sequence (A006255).

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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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A259527 a(n) gives the number of sequences n = b_1 < b_2 < ... < b_t = A006255(n) such that b_1*b_2*...*b_t is a perfect square.

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

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A067565 Inverse of R. L. Graham's sequence (A006255), or zero if and only if n is a prime.

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A269045 Indices k such that A006255(k) != A070229(k); that is, the k-th term of R. L. Graham's sequence is not equal to k + lpf(k).

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A233421 Let m = n-th nonsquare = A000037(n); then a(n) = A006255(m).

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A255167 a(n) = A072905(n) - A006255(n).

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A255363 Numbers with the property that A006255(k) = A070229(k).

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A259527 a(n) gives the number of sequences n = b_1 < b_2 < ... < b_t = A006255(n) such that b_1b_2...*b_t is a perfect square.

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

A280244 Lexicographically ordered list of sequences that meet the criteria for R. L. Graham's sequence: k = a_1 < a_2 < ... < a_t = A006255(k) and a_1a_2...*a_t is a square.