A080508 -id:A080508 - cp's OEIS frontend

A080509 Last term in n-th row of A080508.

1, 4, 4, 216, 324, 6075000, 30375000, 750453558750000, 19699405917187500, 459652804734375000, 9652708899421875000, 578346405423301688948883281250000, 111331683043985575122660031640625000, 77892265302487151682927242578030755976166730468750000

Offset: 1

Amarnath Murthy, Mar 20 2003

nonn

			For n=5, first four terms of row are 1, 2, 3, 4, with product 24 = 2^3*3^1. So last term is 2^(5-3)*3^(5-1) = 2^2*3^4 = 324.

Jeppe Stig Nielsen, Table of n, a(n) for n = 1..54

Cf. A080508.

MapAt[4 # &, Array[Apply[Times, Prime@ Range@ PrimePi[# - 1]]^#/(# - 1)! &, 14], 2] (* Michael De Vlieger, Nov 05 2018 *)

a(n) = {if (n == 1, return (1)); if (n == 2, return (2^2)); f  = factor((n-1)!); prod(i = 1, #f~, f[i,1]^(n - f[i,2]));} \\ Michel Marcus, Aug 30 2013

a(n) = if(n==2, 4, prod(i=1,primepi(n-1),prime(i))^n/(n-1)!) \\ Jeppe Stig Nielsen, Nov 04 2018

For n!=2, a(n) = (A034386(n - 1))^n / (n - 1)!. - Jeppe Stig Nielsen, Nov 04 2018

More terms from Michel Marcus, Aug 30 2013

A186944 Geometric mean of n-th row of A080508.

Original entry on oeis.org

1, 2, 2, 6, 6, 30, 30, 210, 210, 210, 210, 2310, 2310, 30030, 30030, 30030, 30030, 510510, 510510, 9699690, 9699690, 9699690, 9699690, 223092870, 223092870, 223092870, 223092870, 223092870, 223092870, 6469693230, 6469693230, 200560490130, 200560490130

Offset: 1

Michel Marcus, Aug 30 2013

nonn

Cf. A080509, A083907, A034386.

a(n) = {if (n == 1, return (1)); if (n == 2, return (2)); f  = factor((n-1)!); prod(i=1, #f~, f[i,1]);} \\ Michel Marcus, Aug 30 2013

a(n) = if(n==2, 2, prod(i=1,primepi(n-1),prime(i))) \\ Jeppe Stig Nielsen, Nov 04 2018

For n != 2, a(n) = A034386(n-1). - Jeppe Stig Nielsen, Nov 04 2018

A080504 Triangle whose n-th row contains the least set (ordered lexicographically) of n distinct positive integers whose arithmetic and geometric means are both integers.

Original entry on oeis.org

1, 1, 9, 1, 2, 108, 1, 2, 5, 1000, 1, 2, 3, 4, 1012500, 1, 2, 3, 4, 8, 15552, 1, 2, 3, 4, 5, 6, 25015118625000, 1, 2, 3, 4, 5, 6, 11, 17757684573750000, 1, 2, 3, 4, 5, 6, 7, 8, 19699405917187500, 1, 2, 3, 4, 5, 6, 7, 8, 14, 295491088757812500, 1, 2, 3, 4, 5, 6, 7

Offset: 1

Amarnath Murthy, Mar 20 2003

Row n has the form {1,2,...,n-2,x,y} where n-1 <= x < y. x is minimal such that y exists.

			The fourth row contains 1,2,5,1000, with AM=252 and GM=10. There is no set of the form {1,2,3,y} or {1,2,4,y} whose AM and GM are both integers.

Cf. A080505, A081556, A080506, A080507, A080508, A080511.

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 *)

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A080509 Last term in n-th row of A080508.

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A186944 Geometric mean of n-th row of A080508.

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PARI

PARI

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A080504 Triangle whose n-th row contains the least set (ordered lexicographically) of n distinct positive integers whose arithmetic and geometric means are both integers.

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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