A066400 -id:A066400 - cp's OEIS frontend

A277649 Where records occur in A066400.

1, 2, 8, 14, 52, 99, 589, 594, 595, 1566, 1961, 3465, 5301

Offset: 1

David A. Corneth, Oct 25 2016

The respective lengths are 1, 3, 4, ..., 14.
The a(n)-th tuples from A245499 are
[1], [2, 3, 6], [8, 10, 12, 15], [14, 15, 18, 20, 21], [52, 54, 56, 60, 63, 65], [99, 104, 105, 108, 110, 112, 117], [589, 594, 595, 600, 608, 612, 616, 620], [594, 595, 598, 600, 605, 612, 616, 621, 624], [595, 598, 600, 605, 608, 612, 616, 621, 624, 627], [1566, 1568, 1573, 1575, 1581, 1584, 1587, 1595, 1596, 1612, 1615], [1961, 1972, 1974, 1978, 1984, 1989, 1995, 1998, 2001, 2014, 2015, 2021], [3465, 3478, 3480, 3483, 3485, 3496, 3500, 3509, 3515, 3519, 3520, 3525, 3526], [5301, 5304, 5307, 5310, 5313, 5320, 5324, 5332, 5336, 5346, 5355, 5368, 5369, 5375]

Cf. A006255, A066400.

Suggested by Peter Kagey in a comment of A066400, Oct 24 2016.

Name corrected by Omar E. Pol, Nov 06 2016

A006255 R. L. Graham's sequence: a(n) = smallest m for which there is a sequence n = b_1 < b_2 < ... < b_t = m such that b_1b_2...*b_t is a perfect square.

Original entry on oeis.org

1, 6, 8, 4, 10, 12, 14, 15, 9, 18, 22, 20, 26, 21, 24, 16, 34, 27, 38, 30, 28, 33, 46, 32, 25, 39, 35, 40, 58, 42, 62, 45, 44, 51, 48, 36, 74, 57, 52, 50, 82, 56, 86, 55, 60, 69, 94, 54, 49, 63, 68, 65, 106, 70, 66, 72, 76, 87, 118, 75, 122, 93, 77, 64, 78, 80, 134, 85, 92, 84

Offset: 1

N. J. A. Sloane and Robert G. Wilson v

Every nonprime appears exactly once in this sequence.
If n is a square we can take t=1 and a(n) = n. If n is a prime > 3, then a(n) = 2n and t=3. If n is twice a prime, say p, then a(n) = 3p most of the time. The sequence b_1 < b_2 < ... < b_t will not contain either perfect squares or primes for they bring nothing to the solution. Also I know of no n such that t = 2. - Robert G. Wilson v, Jan 30 2002
Let k be a fixed integer and p be a prime, then a(k*p) = (k+1)*p for sufficiently large p. - Peter Kagey, Feb 03 2015
From David A. Corneth, Oct 26 2016: (Start)
Is for all k*p in A277624, a(k*p) = (k+1) * p?
Conjecture: Let b(n) = A006530(A007913(n)). If b(n)^2 >= 2 * n then a(n) = n + b(n) except for n = 3, 10, and 171.
(End)
a(n) <= A072905(n).
a(n) <= 2*n for all n > 3.
a(n) >= n + A006530(A007913(n)) for all nonsquare n. - Peter Kagey, Feb 21 2015

			a(2) = 6 because the best such sequence is 2,3,6.
For n = 3 through 6 the {smallest m then smallest t then smallest product} solutions are 3,6,8; 4; 5,8,10; 6,8,12.

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 2nd. ed., Problem 4.39, pages 147, 616, 533. [Reference revised by N. J. A. Sloane, Jan 13 2014]
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Peter Kagey, Table of n, a(n) for n = 1..10000
Sarosh Adenwalla, On a Generalisation of a Function of Ron Graham's, arXiv:2504.19196 [math.NT], 2025.
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]
Peter Kagey and Krishna Rajesh, On a Conjecture about Ron Graham's Sequence, arXiv:2410.04728 [math.NT], 2024.

Having minimized m, next minimize t, then minimize product: A066400 and A066401 give values of t and square root of b_1*...*b_t.
If squares are omitted we get A233421.
A067565 is the inverse of R. L. Graham's sequence.
Cf. A245499, A070229, A072905, A092487.

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, 40}] (* Michael De Vlieger, Oct 26 2016 *)

If n is a square we can take t=1 and a(n)=n.
a(n) = A245499(n,A066400(n)). - Reinhard Zumkeller, Jul 25 2014
a(n) = A092487(n) + n. - Peter Kagey, Oct 22 2016

More terms from Robert G. Wilson v, Jan 30 2002

Erroneous program (pointed out by Peter Kagey) removed by Reinhard Zumkeller, Nov 28 2014

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.

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

A245530 a(n) = smallest square which is the product of a minimal set of distinct numbers not less than n.

Original entry on oeis.org

1, 36, 144, 4, 400, 576, 784, 14400, 9, 32400, 4356, 3600, 6084, 1587600, 129600, 16, 10404, 11664, 23104, 14400, 15876, 17424, 33856, 20736, 25, 876096, 25401600, 1254400, 53824, 44100, 61504, 57600, 85377600, 4161600, 2822400, 36, 136900, 116964, 97344

Offset: 1

Reinhard Zumkeller, Jul 25 2014

nonn

a(n) mod n = 0; a(n) mod A006255(n) = 0.

Peter Kagey, Table of n, a(n) for n = 1..1000 (First 125 terms from Reinhard Zumkeller)

Cf. A006255, A066400, A066401, A245499.

Haskell
```
a245530 = product . a245499_row
```

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

a(n) = Product_{k=1..A066400(n)} A245499(n,k), product of n-th row in A245499.

a(n) = A066401(n)^2.

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).

A305677 Number of subsets of {n+1, n+2, ..., A072905(n)-1} whose product has the same squarefree part as n.

Original entry on oeis.org

1, 2, 8, 1, 64, 256, 2048, 4, 1, 131072, 262144, 32, 8388608, 33554432, 134217728, 1, 2147483648, 8, 34359738368, 1024, 549755813888, 4398046511104, 17592186044416, 8192, 2, 1125899906842624, 32, 65536, 72057594037927936, 576460752303423488

Offset: 1

Peter Kagey, Jun 08 2018

nonn

Conjecture: a(n) > 0.
If the conjecture is true, all terms are powers of two, and a(n) >= A259527(n).
a(n) = 0 if and only if A066400(n) = 2.
a(n) = 0 if and only if A255167(n) = 0.
a(n) <= 2^(A067722(n) - 1). - Peter Kagey, Nov 13 2018

			For n = 3, the a(3) = 8 subsets of {4, 5, ..., 11} with a product with squarefree part of 3 are {4, 5, 6, 9, 10}, {4, 5, 6, 10}, {4, 6, 8}, {4, 6, 8, 9}, {5, 6, 9, 10}, {5, 6, 10}, {6, 8}, and {6, 8, 9}.

Peter Kagey, Table of n, a(n) for n = 1..200

Cf. A005117, A006255, A066400, A072905, A255167.

A260896 a(n) gives the number of integers m such that there exist k and h with 2n^2 < mk^2 < 2(n+1)^2 and 2n^2 < 2mh^2 < 2(n+1)^2.

Original entry on oeis.org

0, 1, 0, 1, 1, 1, 3, 3, 2, 3, 2, 2, 3, 3, 0, 3, 1, 4, 2, 3, 3, 1, 6, 3, 4, 4, 5, 3, 2, 5, 4, 8, 4, 4, 5, 1, 5, 6, 4, 5, 3, 6, 2, 5, 7, 5, 8, 4, 7, 4, 7, 7, 7, 10

Offset: 0

Peter Kagey, Aug 03 2015

nonn

A072905(2n^2) > A006255(2n^2) and A066400(2n^2) > 2 for all n such that a(n) > 0.

Conjecture: a(n) > 0 for all n > 14.

			For n=12 the a(12)=3 solutions are 3, 6, and 37:
  (1) (a) 2 * 12^2 <      3 * 10^2 < 2 * 13^2
      (b) 2 * 12^2 < 2 *  3 *  7^2 < 2 * 13^2
  (2) (a) 2 * 12^2 <      6 *  7^2 < 2 * 13^2
      (b) 2 * 12^2 < 2 *  6 *  5^2 < 2 * 13^2
  (3) (a) 2 * 12^2 <     37 *  3^2 < 2 * 13^2
      (b) 2 * 12^2 < 2 * 37 *  2^2 < 2 * 13^2

Peter Kagey, Table of n, a(n) for n = 0..10000

Cf. A006255, A066400, A072905.

A278817 The least t such that there exists a sequence n = b_1 < b_2 < ... < b_t = A277278(n) such that b_1 + b_2 +...+ b_t is a perfect square.

Original entry on oeis.org

1, 1, 3, 2, 1, 5, 2, 2, 6, 1, 3, 3, 2, 4, 3, 3, 1, 2, 4, 3, 3, 5, 2, 2, 2, 1, 3, 4, 4, 2, 2, 2, 3, 4, 4, 6, 1, 5, 3, 2, 2, 5, 5, 5, 3, 3, 3, 3, 2, 1, 6, 6, 3, 3, 3, 3, 6, 6, 2, 2, 2, 4, 4, 3, 1, 7, 7, 4, 4, 2, 2, 2, 3, 3, 3, 5, 5, 5, 5, 4, 2, 1, 2, 2, 2, 5, 5

Offset: 0

Peter Kagey, Nov 28 2016

nonn

a(n) = 1 if and only if n is square.

a(n) = 2 if and only if A277278(n) = A278818(n).

			a(0) = 1 via 0                  = 0^2
a(1) = 1 via 1                  = 1^2
a(2) = 3 via 2 + 3 + 4          = 3^2
a(3) = 2 via 3 + 6              = 3^2
a(4) = 1 via 4                  = 2^2
a(5) = 5 via 5 + 6 + 7 + 8 + 10 = 6^2
a(6) = 2 via 6 + 10             = 4^2

Peter Kagey, Table of n, a(n) for n = 0..3000

Cf. A066400, A277278, A278818.

A305709 Least k such that there exists a three-term sequence n = b_1 < b_2 < b_3 = k such that b_1 * b_2 * b_3 is square.

Original entry on oeis.org

8, 6, 8, 16, 10, 12, 14, 18, 25, 20, 22, 20, 26, 24, 27, 32, 34, 27, 38, 30, 28, 33, 46, 32, 48, 52, 40, 45, 58, 42, 62, 45, 48, 54, 56, 64, 74, 57, 52, 50, 82, 56, 86, 55, 60, 69, 94, 54, 72, 63, 75, 78, 106, 75, 90, 72, 76, 96, 118, 80, 122, 96, 84, 98, 104

Offset: 1

Peter Kagey, Jun 08 2018

nonn

a(n) >= A006255(n), and a(n) = A006255(n) if and only if A066400(n) = 3.

Conjecture: a(n) < A072905(n) with finitely many nonsquare exceptions.

			For n = 3 the sequence is 3, 6, 8; so a(3) = 8;
for n = 4 the sequence is 4, 9, 16; so a(4) = 16;
for n = 5 the sequence is 5, 8, 10; so a(5) = 10.

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

Cf. A006255, A066400, A072905.

cp's OEIS Frontend

A277649 Where records occur in A066400.

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A006255 R. L. Graham's sequence: a(n) = smallest m for which there is a sequence n = b_1 < b_2 < ... < b_t = m such that b_1*b_2*...*b_t is a perfect square.

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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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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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A245530 a(n) = smallest square which is the product of a minimal set of distinct numbers not less than n.

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

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A305677 Number of subsets of {n+1, n+2, ..., A072905(n)-1} whose product has the same squarefree part as n.

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A260896 a(n) gives the number of integers m such that there exist k and h with 2n^2 < mk^2 < 2(n+1)^2 and 2n^2 < 2mh^2 < 2(n+1)^2.

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A006255 R. L. Graham's sequence: a(n) = smallest m for which there is a sequence n = b_1 < b_2 < ... < b_t = m 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).