A081456 -id:A081456 - cp's OEIS frontend

A081457 a(n) = A081456(n)^(1/2).

1, 2, 4, 6, 12, 36, 120, 240, 720, 2520, 10080, 30240, 110880, 443520, 1814400, 6652800, 26611200, 133056000, 518918400, 2075673600, 10378368000, 58118860800, 261534873600, 1270312243200, 5928123801600, 29640619008000, 168951528345600, 844757641728000, 4505374089216000, 25342729251840000

Offset: 1

Amarnath Murthy, Mar 21 2003

nonn

Max Alekseyev, Table of n, a(n) for n = 1..50

Cf. A081454, A081455, A081456, A083489.

a(n) equals smallest integer m>0 such that A086435(m^2)=n-1. - Max Alekseyev, Jul 16 2009

More terms from David Garber, Jun 17 2003
More terms from Ray G. Opao, Aug 01 2005
Corrected and extended by R. J. Mathar, Nov 12 2006
More terms from Max Alekseyev, Jun 05, Jul 16 2009
Corrected and extended by Max Alekseyev, Jun 03 2023

A083489 Triangle read by rows where the n-th row gives the reverse-lexicographically earliest sequence of n distinct integers > 1 whose product equals the smallest possible square (=A081456(n)).

Original entry on oeis.org

4, 2, 8, 2, 3, 6, 2, 3, 4, 6, 2, 3, 4, 6, 9, 2, 3, 5, 6, 8, 10, 2, 3, 4, 5, 6, 8, 10, 2, 3, 4, 5, 6, 8, 9, 10, 2, 3, 4, 5, 6, 7, 9, 10, 14, 2, 3, 4, 5, 7, 8, 9, 10, 12, 14, 2, 3, 4, 5, 6, 7, 8, 9, 12, 14, 15, 2, 3, 4, 5, 6, 7, 8, 9, 12, 14, 15, 16, 2, 3, 4, 6, 7, 8, 9, 10, 12, 14, 15, 16, 18, 2

Offset: 1

Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), May 03 2003

			4
2 8
2 3 6
2 3 4 6
2 3 4 6 9
2 3 5 6 8 10
...

Cf. A081454, A086435, A081457.

Results from the triangle in A081454 by removing first column (equivalently, by removing all 1's from the sequence).

Corrected by R. J. Mathar, Jul 18 2007

Corrected and extended by Max Alekseyev, Jul 16 2009

A081454 Triangle read by rows in which the n-th row contains n distinct numbers whose product is a square, which is minimal over all choices for n distinct numbers.

Original entry on oeis.org

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

Offset: 1

Amarnath Murthy, Mar 21 2003

In case there is more than one solution, choose the one where the maximal number is minimal.

			Triangle begins:
  1;
  1,  4;
  1,  2,  8;
  1,  2,  3,  6;
  1,  2,  3,  4,  6;
  1,  2,  3,  4,  6,  9;
  1,  2,  3,  5,  6,  8, 10;
  1,  2,  3,  4,  5,  6,  8, 10;
  1,  2,  3,  4,  5,  6,  8,  9, 10;
  ...
The 7th row could also be 1, 2, 3, 4, 5, 8, 15, but this has a larger last term.

Max Alekseyev, Rows n = 1..50, flattened

Cf. A081455, A081456, A081457.

A081454aux := proc(n,s,mfact) local d,findx,f ; if n = 1 then if s <= mfact then RETURN([s]) ; else RETURN([]) ; fi ; else d := numtheory[divisors](s) ; for findx from n to nops(d) do if op(findx,d) <= mfact then f := A081454aux(n-1,s/op(findx,d),op(findx,d)-1) ; if nops(f) <> 0 then RETURN([op(f),op(findx,d)]) ; fi ; fi ; od ; RETURN([]) ; fi ; end: A081454row := proc(n) local p,s,d,findx,f ; p :=1 ; s :=1 ; while true do d := numtheory[divisors](s) ; if nops(d) >= n then if n = 1 then RETURN([1]) ; else for findx from n to nops(d) do f := A081454aux(n-1,s/op(findx,d),op(findx,d)-1) ; if nops(f) <> 0 then RETURN([op(f),op(findx,d)]) ; fi ; od; fi ; fi ; p := p+1 ; s := p^2 ; od ; end: for n from 1 to 14 do r := A081454row(n) : for i from 1 to n do printf("%d,",op(i,r) ) ; od ; od : # R. J. Mathar, Nov 12 2006

T[n_] := T[n] = SortBy[MinimalBy[Select[Subsets[Range[2n+2], {n}], #[[1]] == 1 && IntegerQ@Sqrt[Times @@ #]&], Times @@ #&], Last] // First;
Table[Print[n, " ", T[n]]; T[n], {n, 1, 12}] // Flatten (* Jean-François Alcover, Jun 03 2023 *)

Edited and extended by David Garber, Jun 17 2003
More terms from Ray G. Opao, Aug 01 2005
Corrected and extended by R. J. Mathar, Nov 12 2006
More terms from Max Alekseyev, Apr 25 2009
Correct row #13 conjectured by Jean-François Alcover and confirmed by Max Alekseyev, Jun 03 2023

A282193 a(n) is the minimal product of a positive integer sequence of length n with no duplicate substrings (forward or backward) of length greater than 1.

Original entry on oeis.org

1, 1, 2, 4, 6, 12, 36, 96, 240, 480, 1440, 5760, 17280, 40320, 120960, 483840, 1935360, 5806080, 17418240, 69672960, 348364800, 1045094400, 3832012800, 15328051200, 76640256000, 229920768000, 919683072000, 4598415360000, 22072393728000, 71735279616000, 286941118464000, 1434705592320000

Offset: 1

Peter Kagey, Feb 08 2017

nonn

			a(1)  = 1    via [1];
a(2)  = 1    via [1,1];
a(3)  = 2    via [1,1,2];
a(4)  = 4    via [1,1,2,2];
a(5)  = 6    via [1,1,2,3,1];
a(6)  = 12   via [1,1,2,2,3,1];
a(7)  = 36   via [1,1,2,2,3,3,1];
a(8)  = 96   via [1,1,2,2,3,1,4,2];
a(9)  = 240  via [1,1,2,2,3,1,4,5,1];
a(10) = 480  via [1,1,2,2,3,1,4,2,5,1];
a(11) = 1440 via [1,1,2,2,3,3,1,4,2,5,1];
a(12) = 5760 via [1,1,2,2,3,1,4,2,5,1,6,2].
...
[1,2,3,1,2] is invalid because the substring [1,2] appears twice.
[1,2,1] is invalid because the substring [1,2] appears twice (once forward and once backward).

Peter Kagey, Haskell program for A282193.

Cf. A282168 is the sum analog.

Cf. A081456, A282164, A282169, A282170.

Terms a(13) onward from Max Alekseyev, Feb 05 2025

A282169 a(n) is the minimal product of a positive integer sequence of length n with no duplicate substrings of length greater than 1, and every number different from its neighbors.

Original entry on oeis.org

1, 2, 2, 6, 6, 24, 24, 120, 120, 576, 720, 2880, 4320, 17280, 30240, 120960, 241920, 967680, 1935360, 8709120, 17418240, 87091200, 174182400, 870912000, 1741824000, 9580032000, 19160064000, 104509440000, 229920768000, 1149603840000, 2759049216000, 13795246080000, 33108590592000, 165542952960000, 430411677696000

Offset: 1

Peter Kagey, Feb 07 2017

nonn

			[1,1] is not a valid sequence because 1 is self-adjacent.
[1,2,3,1,2] is not valid because the substring [1,2] appears twice.
  a(1)  = 1   via [1];
  a(2)  = 2   via [1,2];
  a(3)  = 2   via [1,2,1];
  a(4)  = 6   via [1,2,1,3];
  a(5)  = 6   via [1,2,1,3,1];
  a(6)  = 24  via [1,2,1,3,1,4];
  a(7)  = 24  via [1,2,1,3,1,4,1];
  a(8)  = 120 via [1,2,1,3,1,4,1,5];
  a(9)  = 120 via [1,2,1,3,1,4,1,5,1];
  a(10) = 576 via [1,2,1,3,1,4,2,3,4,1];
  a(11) = 720 via [1,2,1,3,1,4,1,5,1,6,1].

Cf. A081456, A282164, A282193, A282170, A284434.

Cf. A282166 is the sum analog.

Terms a(12) onward from Max Alekseyev, Feb 04 2025

A282170 a(n) is the minimal product of a positive integer sequence of length n with no duplicate substrings (forward or backward) of length greater than 1, and no self-adjacent terms.

Original entry on oeis.org

1, 2, 6, 6, 24, 48, 120, 240, 1440, 2880, 10080, 20160, 120960, 322560, 1209600, 2903040, 17418240, 58060800, 174182400, 638668800, 3483648000, 15328051200, 38320128000, 199264665600, 919683072000, 4828336128000, 11955879936000, 71735279616000, 334764638208000, 1506440871936000, 5021469573120000, 30128817438720000

Offset: 1

Peter Kagey, Feb 07 2017

nonn

			  a(1)  = 1     via [1];
  a(2)  = 2     via [1,2];
  a(3)  = 6     via [1,2,3];
  a(4)  = 6     via [1,2,3,1];
  a(5)  = 24    via [1,2,3,1,4];
  a(6)  = 48    via [1,2,3,1,4,2];
  a(7)  = 120   via [1,2,3,1,4,5,1];
  a(8)  = 240   via [1,2,3,1,4,2,5,1];
  a(9)  = 1440  via [1,2,3,1,4,2,5,1,6];
  a(10) = 2880  via [1,2,3,1,4,2,5,1,6,2];
  a(11) = 10080 via [1,2,3,1,4,2,5,1,6,7,1].
Examples:
  [1,1] is invalid because 1 is self-adjacent.
  [1,2,3,1,2] is invalid because the substring [1,2] appears twice.
  [1,2,1] is invalid because the substring [1,2] appears twice (once forward and once backward).

Cf. A081456, A282164, A282169, A282170, A282193.

Cf. A282167 is the sum analog.

Terms a(12) onward from Max Alekseyev, Feb 04 2025

A081455 Final entry of n-th row of triangle in A081454.

Original entry on oeis.org

1, 4, 8, 6, 6, 9, 10, 10, 10, 14, 14, 15, 22, 22, 20, 22, 22, 25, 26, 26, 26, 28, 28, 34, 34, 34, 38, 38, 38, 38, 38, 40, 46, 46, 46, 46, 46, 50, 58, 58, 58, 58, 58, 62, 62, 58, 62, 62, 62, 74

Offset: 1

Amarnath Murthy, Mar 21 2003

Cf. A081454, A081456, A081457.

More terms from David Garber, Jun 17 2003
Corrected and extended by R. J. Mathar, Nov 12 2006
More terms from Max Alekseyev, Jul 16 2009
Corrected and extended by Max Alekseyev, Jun 03 2023

cp's OEIS Frontend

A081457 a(n) = A081456(n)^(1/2).

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A083489 Triangle read by rows where the n-th row gives the reverse-lexicographically earliest sequence of n distinct integers > 1 whose product equals the smallest possible square (=A081456(n)).

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A081454 Triangle read by rows in which the n-th row contains n distinct numbers whose product is a square, which is minimal over all choices for n distinct numbers.

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A282193 a(n) is the minimal product of a positive integer sequence of length n with no duplicate substrings (forward or backward) of length greater than 1.

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A282169 a(n) is the minimal product of a positive integer sequence of length n with no duplicate substrings of length greater than 1, and every number different from its neighbors.

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A282170 a(n) is the minimal product of a positive integer sequence of length n with no duplicate substrings (forward or backward) of length greater than 1, and no self-adjacent terms.

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A081455 Final entry of n-th row of triangle in A081454.

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