A096123 -id:A096123 - cp's OEIS frontend

A096974 Numbers n such that A096123(n) = n!/(nextprime(n/2)-1)!.

1, 2, 3, 5, 8, 9, 11, 12, 13, 20, 21, 24, 25, 32, 33, 36, 37, 44, 45, 54, 55, 56, 57, 60, 61, 72, 73, 80, 81, 83, 84, 85, 92, 93, 104, 105, 116, 117, 120, 121, 130, 131, 132, 133, 140, 141, 144, 145, 156, 157, 164, 165, 176, 177, 192, 193, 200, 201, 204, 205, 212, 213

Offset: 1

Klaus Brockhaus, Jul 17 2004

nonn

Suggested by the apparently false conjecture (A. Murthy) that A096123(n) = n!/(nextprime(n/2)-1)! for all sufficiently large n.

Observation: If n is an even term then n+1 is also a term. Odd terms n that are not preceded by term n-1 are very rare; only 1, 5, 11, 83, 455, 623, 839, 1139, 1199, 2039, 2459, 2579, 2639, 2855, 2975 have been found up to 3000.

			A096123(11) = 11!/(nextprime(11/2)-1)! = 11!/(6-1)! = 55440, hence 11 is a term.

Cf. A096123.

{for(n=1,215,p=1;k=0;b=1;while(b&&k

A056042 a(n) = n!/(k!)^2, where k is the largest number such that (k!)^2 divides n!.

Original entry on oeis.org

1, 2, 6, 6, 30, 20, 140, 70, 630, 7, 77, 924, 12012, 3432, 51480, 12870, 218790, 48620, 923780, 184756, 3879876, 705432, 16224936, 2704156, 67603900, 10400600, 280816200, 178296, 5170584, 155117520, 4808643120, 601080390, 19835652870

Offset: 1

Labos Elemer, Jul 25 2000

nonn

Least integer of the form n!/{(n-k)!}^2.

Similar to but different from A001405.

			E.g. for n=9, 10, 11, 12, a(n)=630, 7, 77, 924 while the corresponding central binomial coefficients are 126, 252, 462, 924 respectively.

Ivan Neretin, Table of n, a(n) for n = 1..1000

Cf. A000142, A001405, A000188, A055772, A096123, A096125.

f[n_] := Min[ Select[ Table[ n!/(n - k)!^2, {k, n}], IntegerQ[ # ] &]]; Table[ f[n], {n, 33}] (Robert G. Wilson v)

A096125 Least value of k such that n!/((n-k)!)^2 is an integer.

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 4, 4, 5, 4, 5, 6, 7, 7, 8, 8, 9, 9, 10, 10, 11, 11, 12, 12, 13, 13, 14, 13, 14, 15, 16, 16, 17, 17, 18, 18, 19, 19, 20, 20, 21, 21, 22, 22, 23, 23, 24, 24, 25, 25, 26, 26, 27, 26, 27, 28, 29, 29, 30, 30, 31, 31, 32, 32, 33, 33, 34, 34, 35, 35, 36, 36, 37, 37, 38

Offset: 1

Amarnath Murthy, Jul 01 2004

nonn

If p is the first prime > n/2, then a(n) > n-p. - Robert Israel, Jun 27 2018

			a(10) = 4.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A096123, A096124.

f:= proc(n) local p,k;
  p:= nextprime(floor(n/2));
  for k from n-p+1 do
    if (n!/((n-k)!)^2)::integer then return k fi
  od
end proc:
f(1):= 1:
map(f, [$1..100]); # Robert Israel, Jun 27 2018

f[n_] := Block[{i = 1, t = Table[n!/(n - k)!^2, {k, n}]}, While[ !IntegerQ[ t[[i]]], i++ ]; i]; Table[ f[n], {n, 75}] (* Robert G. Wilson v, Jul 03 2004 *)

Edited, corrected and extended by Robert G. Wilson v, Jul 03 2004

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A096974 Numbers n such that A096123(n) = n!/(nextprime(n/2)-1)!.

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A056042 a(n) = n!/(k!)^2, where k is the largest number such that (k!)^2 divides n!.

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A096125 Least value of k such that n!/((n-k)!)^2 is an integer.

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