A081457 -id:A081457 - cp's OEIS frontend

A081456 Product of entries in n-th row of triangle in A081454.

1, 4, 16, 36, 144, 1296, 14400, 57600, 518400, 6350400, 101606400, 914457600, 12294374400, 196709990400, 3292047360000, 44259747840000, 708155965440000, 17703899136000000, 269276305858560000, 4308420893736960000, 107710522343424000000, 3377801980689776640000

Offset: 1

Amarnath Murthy, Mar 21 2003

nonn

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

Cf. A081454, A081455, A081457.

a(n) equals smallest integer square m^2 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

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

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

A256881 a(n) = n!/ceiling(n/2).

Original entry on oeis.org

1, 2, 3, 12, 40, 240, 1260, 10080, 72576, 725760, 6652800, 79833600, 889574400, 12454041600, 163459296000, 2615348736000, 39520825344000, 711374856192000, 12164510040883200, 243290200817664000, 4644631106519040000, 102181884343418880000, 2154334728240414720000

Offset: 1

M. F. Hasler, Apr 22 2015

nonn

Original name was: n!/round(n/2). - Robert Israel, Sep 03 2018

Robert Israel, Table of n, a(n) for n = 1..450
Pierre-Alain Sallard, Sum of repeated integrals of sinh.

Cf. A009445, A052612, A052616, A052849, A081457, A208529, A098558, A107991, A110468, A229244, A256031.

[Factorial(n)/Round(n/2): n in [1..30]]; // Vincenzo Librandi, Apr 23 2015

Maple
```
A256881 := n!/round(n/2);
```

Function[x, 1/x] /@
CoefficientList[Series[(Sinh[x] + x*Exp[x])/2, {x, 0, 20}], x] (* Pierre-Alain Sallard, Dec 15 2018 *)

PARI
```
A256881(n)=n!/round(n/2)
```

a(2n) = 2*A009445(n) = A052612(2n-1) = A052616(2n-1) = A052849(2n-1) = A098558(2n-1) = A081457(3n-1) = A208529(2n+1) = A256031(2n-1).
a(2n+1) = A110468(n) = A107991(2n+2) = A229244(2n+1), n>=0.
From Robert Israel, Sep 03 2018: (Start)
E.g.f.: -(1+1/x)*log(1-x^2).
n*(n+1)*(n+2)*a(n)+(n+2)*a(n+1)-(n+3)*a(n+2)=0. (End)
a(n) = 2/([x^n](sinh(x) + x*exp(x))). - Pierre-Alain Sallard, Dec 15 2018
Sum_{n>=1} 1/a(n) = (3*e-1/e)/4 = (e + sinh(1))/2. - Amiram Eldar, Feb 02 2023

Definition clarified by Robert Israel, Sep 03 2018

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

cp's OEIS Frontend

A081456 Product of entries in n-th row of triangle in A081454.

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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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Maple

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

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Author

Keywords

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Extensions

A256881 a(n) = n!/ceiling(n/2).

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Author

Keywords

Comments

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Programs

Magma

Maple

Mathematica

PARI

Formula

Extensions

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)).

Views

Author

Keywords

Examples

Crossrefs

Formula

Extensions