A245499 -id:A245499 - cp's OEIS frontend

A277606 Frequency of n in A245499.

1, 1, 2, 1, 1, 3, 1, 5, 1, 3, 1, 4, 1, 2, 5, 1, 1, 7, 1, 4, 2, 2, 1, 5, 1, 2, 5, 3, 1, 3, 1, 9, 2, 2, 5, 1, 1, 2, 2, 5, 1, 4, 1, 2, 4, 2, 1, 6, 1, 8, 2, 3, 1, 5, 2, 5, 2, 2, 1, 6, 1, 2, 7, 1, 3, 4, 1, 2, 2, 5, 1, 13, 1, 2, 10, 2, 6, 2, 1, 7, 1, 2

Offset: 1

David A. Corneth, Oct 23 2016

nonn

From David A. Corneth, Jan 02 2018: (Start)
For every positive integer n, we can create a tuple b of t increasing positive integers with b_1 = n, the product of these elements a perfect square and the largest element as small as possible.
A006255 lists b_t, the largest element of these tuples, A245499 lists these tuples and this sequence lists the frequency of n occurring in such a tuple, i.e., the frequency of n in A245499. (End)
Records occur for n = 1, 3, 6, 8, 18, 32, 72, 200, ... where a(n) is 1, 2, 3, 5, 7, 9, 13, 19, ... respectively.

			8 occurs in rows 3, 5, 6, 7 and 8 being respectively [3, 6, 8], [5, 8, 10], [6, 8, 12], [7, 8, 14] and [8, 10, 12, 15]. These are 5 rows so a(8) = 5.

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A006255, A092487, A245499.

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

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

Original entry on oeis.org

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

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

A245508 Smallest double square (cf. A001105) greater than n-th prime.

Original entry on oeis.org

2, 8, 8, 8, 18, 18, 18, 32, 32, 32, 32, 50, 50, 50, 50, 72, 72, 72, 72, 72, 98, 98, 98, 98, 98, 128, 128, 128, 128, 128, 128, 162, 162, 162, 162, 162, 162, 200, 200, 200, 200, 200, 200, 200, 200, 200, 242, 242, 242, 242, 242, 242, 242, 288, 288, 288, 288

Offset: 1

Reinhard Zumkeller, Jul 25 2014

nonn

For n > 2: prime(n) < a(n) < 2*prime(n) and a(n) = A245499(A000040(n),2).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A000040, A001105, A145445.

import Data.List (genericIndex)
a245508 n = genericIndex a245508_list (n-1)
a245508_list = f a000040_list a001105_list where
   f ps'@ (p:ps) xs'@(x:xs) = if p <= x then x : f ps xs' else f ps' xs

Module[{nn=60,ds},ds=2 Range[0,Ceiling[Sqrt[Prime[nn]]]]^2;Join[ {2},Table[ SelectFirst[ds,#>Prime[n]&],{n,2,nn}]]] (* Harvey P. Dale, Jan 07 2020 *)

a(n) = my(k=prime(n)+(n!=1)); while (! issquare(k/2), k+=2); k; \\ Michel Marcus, Jan 24 2022

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.

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 *)

A277649 Where records occur in A066400.

Original entry on oeis.org

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

cp's OEIS Frontend

A277606 Frequency of n in A245499.

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

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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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A245508 Smallest double square (cf. A001105) greater than n-th prime.

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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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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_1*a_2*...*a_t is a square.

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Mathematica

A277649 Where records occur in A066400.

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).

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.