A080504 -id:A080504 - cp's OEIS frontend

A081556 Next-to-last term in n-th row of A080504.

1, 2, 5, 4, 8, 6, 11, 8, 14, 10, 17, 12, 20, 14, 23, 16, 26, 18, 29, 20, 32, 22, 32, 24, 38, 26, 41, 28, 44, 30, 47, 32, 50, 34, 53, 36, 56, 38, 59, 40, 62, 42, 65, 44, 68, 46, 71, 48, 74, 50, 77, 52, 80, 54, 83, 56, 86, 58, 89, 60, 92, 62, 95, 64, 98, 66, 101, 68, 104, 70

Offset: 2

Dean Hickerson, Mar 21 2003

nonn

If n is odd, a(n) = n-1. If n is even, n-1 <= a(n) <= 3n/2-1. For even n up to 100, a(n) = 3n/2-1 unless n = 2 or 24; are there other exceptions? - Dean Hickerson, Mar 21 2003

			a(24)=32 since the 24th row of A080504 can be exhibited as: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 32, y with y = 3^15*5^20*7^21*11^22*13^23*17^23*19^23.

Cf. A080504, A080505, A080506, A080507.

A080505 Last term in n-th row of A080504.

Original entry on oeis.org

1, 9, 108, 1000, 1012500, 15552, 25015118625000, 17757684573750000, 19699405917187500, 295491088757812500, 2754030502015722328328015625000, 69471494082638078965978431093750000

Offset: 1

Amarnath Murthy, Mar 20 2003

nonn

Cf. A080504, A081556, A080506, A080507.

A080506 Geometric mean of n-th row of A080504.

Original entry on oeis.org

1, 3, 6, 10, 30, 12, 210, 330, 210, 210, 2310, 3570, 30030, 2310, 30030, 690690, 510510, 30030, 9699690, 14804790, 9699690, 19399380, 223092870, 9699690, 223092870, 223092870, 223092870, 9146807670, 6469693230, 223092870

Offset: 1

Amarnath Murthy, Mar 20 2003

nonn

Cf. A080504, A080505, A081556, A080507.

A080507 Arithmetic mean of n-th row of A080504.

Original entry on oeis.org

1, 5, 37, 252, 202502, 2595, 3573588375003, 2219710571718754, 2188822879687504, 29549108875781255, 250366409274156575302546875005, 5789291173553173247164869257812506

Offset: 1

Amarnath Murthy, Mar 20 2003

nonn

Cf. A080504, A080505, A081556, A080506.

A080508 Triangle whose n-th row contains the least set (ordered lexicographically) of n distinct positive integers whose geometric mean is an integer.

Original entry on oeis.org

1, 1, 4, 1, 2, 4, 1, 2, 3, 216, 1, 2, 3, 4, 324, 1, 2, 3, 4, 5, 6075000, 1, 2, 3, 4, 5, 6, 30375000, 1, 2, 3, 4, 5, 6, 7, 750453558750000, 1, 2, 3, 4, 5, 6, 7, 8, 19699405917187500, 1, 2, 3, 4, 5, 6, 7, 8, 9, 459652804734375000, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 9652708899421875000

Offset: 1

Amarnath Murthy, Mar 20 2003

The n-th row has the form {1,2,...,n-1,x}, where x is as small as possible.

			Triangle begins:
  1;
  1, 4;
  1, 2, 4;
  1, 2, 3, 216;
  1, 2, 3,   4, 324;
  1, 2, 3,   4,   5, 6075000;
  ...

Robert Israel, Table of n, a(n) for n = 1..1485 (rows 1 to 54, flattened)

Cf. A080509, A080510, A080504, A080511.

f:= proc(n) local F;
  F:= ifactors((n-1)!)[2];
  mul(t[1]^(n-(t[2] mod n)),t=F)
end proc:
f(2):= 4:
seq(op([seq(j,j=1..i-1),f(i)]),i=1..20); # Robert Israel, Nov 04 2018

MapAt[{First@ #, 4 Last@ #} &, Array[Append[Range[# - 1], Apply[Times, Prime@ Range@ PrimePi[# - 1]]^#/(# - 1)!] &, 11], 2] // Flatten (* Michael De Vlieger, Nov 05 2018 *)

More terms using A080509 from Michel Marcus, Nov 04 2018

A080511 Triangle whose n-th row contains the least set (ordered lexicographically) of n distinct positive integers whose arithmetic mean is an integer.

Original entry on oeis.org

1, 1, 3, 1, 2, 3, 1, 2, 3, 6, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 9, 1, 2, 3, 4, 5, 6, 7, 1, 2, 3, 4, 5, 6, 7, 12, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 3, 4, 5, 6, 7, 8, 9, 15, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 18, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 21

Offset: 1

Amarnath Murthy, Mar 20 2003

The n-th row is {1,2,...,n-1,x}, where x=n if n is odd, x=3n/2 if n is even.

			Triangle starts:
1;
1, 3;
1, 2, 3;
1, 2, 3, 6;
1, 2, 3, 4, 5;
1, 2, 3, 4, 5, 9;
1, 2, 3, 4, 5, 6, 7;
1, 2, 3, 4, 5, 6, 7, 12;
...

Cf. A080512, A008619, A080504, A080508.

T:= proc(n) $1..n-1, `if`(irem(n, 2)=1, n, 3*n/2) end:
seq(T(n), n=1..20);  # Alois P. Heinz, Aug 29 2013

row[n_] := Append[Range[n - 1], If[OddQ[n], n, 3 n/2]];
Table[row[n], {n, 1, 20}] // Flatten (* Jean-François Alcover, May 21 2016 *)

A332092 Decimal expansion of Arithmetic-geometric mean AGM(1, 2, 2) defined as limit of the sequence x(n+1) = P(x(n)) with x(0) = (1, 2, 2) and P(a,b,c) = ((a + b + c)/3, sqrt((ab + ac + bc)/3), (abc)^(1/3)).

Original entry on oeis.org

1, 6, 2, 8, 8, 5, 8, 0, 8, 8, 8, 4, 4, 9, 3, 8, 8, 4, 0, 7, 7, 6, 2, 9, 0, 2, 7, 7, 9, 8, 8, 7, 0, 8, 0, 4, 7, 6, 5, 7, 6, 3, 7, 5, 2, 8, 3, 3, 6, 2, 6, 9, 0, 3, 6, 4, 7, 6, 0, 3, 4, 7, 8, 8, 3, 6, 7, 3, 5, 9, 6, 6, 2, 2, 2, 9, 8, 9, 4, 8, 9, 1, 1, 9, 9, 0, 8, 5, 3, 5, 7, 5, 0, 2, 6, 0, 1, 4, 3, 1, 5

Offset: 1

M. F. Hasler, Sep 18 2020

The Arithmetic-geometric mean of two values, AGM(x,y), is the limit of the sequence defined by iterations of (x,y) -> ((x+y)/2, sqrt(xy)). This can be generalized to any number of m variables by taking the vector of the k-th roots of the normalized k-th elementary symmetric polynomials in these variables, i.e., the average of all products of k among these m variables, with k = 1 .. m. After each iteration these m components are in strictly decreasing order unless they are all equal. Once they are in this order, the first one is strictly decreasing, the last one is strictly increasing, therefore they must all have the same limit.

Has this multi-variable AGM already been studied somewhere? Any contributions in that sense are welcome. (Other generalizations have also been proposed, cf. comments on StackExchange.)

			1.62885808884493884077629027798870804765763752833626903647603...

User Mathlover, To find the limit of three terms mean iteration, math.StackExchange.com, Jul 12 2013.
Wikipedia, Arithmetic-geometric mean, created Jan 2, 2002.
Wikipedia, Elementary symmetric polynomial, created Jan 28, 2005.

Cf. other sequences related to the AGM (of two numbers): A061979, A080504, A090852 ff, A127758 ff.

f(k,x,S)={forvec(i=vector(k,i,[1,#x]), S+=vecprod(vecextract(x,i)),2); S/binomial(#x,k)} \\ normalized k-th elementary symmetric polynomial in x
AGM(x)={until(x[1]<=x[#x],x=[sqrtn(f(k,x),k)|k<-[1..#x]]);vecsum(x)/#x}
default(realprecision,100);digits(AGM([1,2,2])\.1^100)

A332093 Decimal expansion of Arithmetic-geometric mean AGM(1, 2, 3) defined as limit of the sequence x(n+1) = P(x(n)) with x(0) = (1, 2, 3) and P(a,b,c) = ((a + b + c)/3, sqrt((ab + ac + bc)/3), (abc)^(1/3)).

Original entry on oeis.org

1, 9, 0, 9, 9, 2, 6, 2, 3, 3, 5, 4, 0, 8, 1, 5, 3, 2, 3, 7, 2, 2, 6, 7, 5, 1, 0, 9, 7, 8, 7, 5, 3, 3, 5, 5, 9, 1, 3, 5, 6, 2, 4, 4, 0, 8, 0, 2, 7, 2, 8, 4, 0, 5, 8, 3, 3, 8, 8, 5, 5, 5, 6, 8, 6, 6, 0, 2, 6, 6, 2, 8, 7, 1, 3, 2, 4, 5, 7, 9, 5, 1, 2, 7, 9, 9, 6, 1, 6, 7, 6, 1, 7, 5, 6, 4, 9, 8, 3, 2, 6

Offset: 1

M. F. Hasler, Sep 18 2020

The Arithmetic-geometric mean of two values, AGM(x,y), is the limit of the sequence defined by iterations of (x,y) -> ((x+y)/2, sqrt(xy)). This can be generalized to any number of m variables by taking the vector of the k-th roots of the normalized k-th elementary symmetric polynomials in these variables, i.e., the average of all products of k among these m variables, with k = 1 .. m. After each iteration these m components are in strictly decreasing order unless they are all equal. Once they are in this order, the first one is strictly decreasing, the last one is strictly increasing, therefore they both converge, and their limits (thus that of all components) must be the same.
Has this multi-variable AGM already been studied somewhere? Any references in that sense or formulas are welcome.
Other 3-argument generalizations of the AGM have been proposed, which all give different values whenever the three arguments are not all equal: replacing P(a,b,c) by (agm(a,b), agm(b,c), agm(a,c)) or (agm(a,agm(b,c)), cyclic...) one gets 1.9091574... resp. 1.9091504..., but these are less straightforwardly generalized to a symmetric function in more than 3 arguments. Using the average of the k-th roots rather than the root of the average (normalized elementary symmetric polynomial) yields 1.89321.... See the two StackExchange links and discussion on the math-fun list. [Edited by M. F. Hasler, Sep 23 2020]

			1.90992623354081532372267510978753355913562440802728405833885556866...

Brad Klee, Iterated averaging of triples, math-fun list (available for subscribers), Sep 18 2020.
User Mathlover, To find the limit of three terms mean iteration, math.StackExchange.com, Jul 12 2013.
Vladimir Reshetnikov, Arithmetic-geometric mean of 3 numbers, math.StackExchange.com, May 22 2016.
Wikipedia, Arithmetic-geometric mean, created Jan 2, 2002.
Wikipedia, Elementary symmetric polynomial, created Jan 28, 2005.

Cf. A332091 = AGM(1,1,2), A332092 = AGM(1,2,2).

Cf. other sequences related to the AGM (of two numbers): A061979, A080504, A090852 ff, A127758 ff.

f(k,x,S)={forvec(i=vector(k,i,[1,#x]), S+=vecprod(vecextract(x,i)),2); S/binomial(#x,k)} \\ normalized k-th elementary symmetric polynomial in x
AGM(x)={until(x[1]<=x[#x],x=[sqrtn(f(k,x),k)|k<-[1..#x]]);vecsum(x)/#x}
default(realprecision,100);digits(AGM([1,2,3])\.1^100)

A332091 Decimal expansion of the arithmetic-geometric mean AGM(1, 1, 2) defined as limit of the sequence x(n+1) = P(x(n)) with x(0) = (1, 1, 2) and P(a,b,c) = ((a + b + c)/3, sqrt((ab + ac + bc)/3), (abc)^(1/3)).

Original entry on oeis.org

1, 2, 9, 4, 5, 7, 5, 1, 0, 8, 1, 1, 6, 6, 1, 2, 6, 4, 3, 4, 4, 8, 6, 4, 3, 4, 9, 8, 2, 1, 0, 0, 3, 5, 3, 6, 7, 4, 0, 3, 7, 9, 7, 2, 7, 2, 1, 5, 6, 4, 2, 4, 5, 8, 6, 8, 0, 8, 6, 6, 4, 1, 7, 2, 3, 9, 5, 6, 5, 9, 8, 7, 4, 8, 5, 8, 9, 6, 2, 0, 5, 9, 7, 5, 6, 5, 9, 8, 7, 6, 7, 6, 7, 1, 4, 2, 5, 6, 4, 7, 4

Offset: 1

M. F. Hasler, Sep 18 2020

See the main entry A332093 for more information on the multi-argument AGM(...) used here. One main motivation for these entries is to find exact formulas for this function which seems not yet well studied in the literature, or at least for particular values like this one, A332092 = AGM(1,2,2) and A332093 = AGM(1,2,3). Any references to possibly existing works using this definition would be welcome.

Other 3-argument generalizations of the AGM have been proposed (cf. A332093) which will give different values for AGM(1,1,2).

			1.294575108116612643448643498210035367403797272156424586808664172...

Vladimir Reshetnikov, Arithmetic-geometric mean of 3 numbers, math.StackExchange.com, May 22 2016.
User Mathlover, To find the limit of three terms mean iteration, math.StackExchange.com, Jul 12 2013.
Wikipedia, Arithmetic-geometric mean, created Jan 2, 2002.

Cf. A332092 (AGM(1,2,2)), A332093 (AGM(1,2,3)).

Cf. other sequences related to the AGM (of two numbers): A061979, A080504, A090852 ff, A127758 ff.

f(k,x,S)={forvec(i=vector(k,i,[1,#x]), S+=vecprod(vecextract(x,i)),2); S/binomial(#x,k)} \\ normalized k-th elementary symmetric polynomial in x
AGM(x)={until(x[1]<=x[#x],x=[sqrtn(f(k,x),k)|k<-[1..#x]]);vecsum(x)/#x}
default(realprecision,100);digits(AGM([1,1,2])\.1^100)

cp's OEIS Frontend

A081556 Next-to-last term in n-th row of A080504.

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A080505 Last term in n-th row of A080504.

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A080506 Geometric mean of n-th row of A080504.

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A080507 Arithmetic mean of n-th row of A080504.

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A080508 Triangle whose n-th row contains the least set (ordered lexicographically) of n distinct positive integers whose geometric mean is an integer.

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A080511 Triangle whose n-th row contains the least set (ordered lexicographically) of n distinct positive integers whose arithmetic mean is an integer.

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Mathematica

A332092 Decimal expansion of Arithmetic-geometric mean AGM(1, 2, 2) defined as limit of the sequence x(n+1) = P(x(n)) with x(0) = (1, 2, 2) and P(a,b,c) = ((a + b + c)/3, sqrt((ab + ac + bc)/3), (abc)^(1/3)).

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PARI

A332093 Decimal expansion of Arithmetic-geometric mean AGM(1, 2, 3) defined as limit of the sequence x(n+1) = P(x(n)) with x(0) = (1, 2, 3) and P(a,b,c) = ((a + b + c)/3, sqrt((ab + ac + bc)/3), (abc)^(1/3)).

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A332091 Decimal expansion of the arithmetic-geometric mean AGM(1, 1, 2) defined as limit of the sequence x(n+1) = P(x(n)) with x(0) = (1, 1, 2) and P(a,b,c) = ((a + b + c)/3, sqrt((ab + ac + bc)/3), (abc)^(1/3)).

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PARI