A075055 -id:A075055 - cp's OEIS frontend

A058008 Numbers k such that (2*k - 1)!/(k!)^2 is an integer.

1, 6, 15, 28, 42, 45, 66, 77, 91, 110, 126, 140, 153, 156, 170, 187, 190, 204, 209, 210, 220, 228, 231, 238, 266, 276, 299, 308, 312, 315, 322, 325, 330, 345, 378, 414, 420, 429, 435, 440, 442, 450, 459, 460, 468, 476, 483, 493, 496, 510, 527, 551, 558, 561, 570

Offset: 1

Labos Elemer, Nov 13 2000

Original name was: Numbers n such that gcd(2*n,C(2*n,n))=2*n.
For n a prime power (see A000961) we have gcd(2*n,C(2*n,n))=2. - Arkadiusz Wesolowski, Jul 01 2012
Also, positions where A075055 differs from A000984. - Ralf Stephan, Dec 11 2004
From Peter Bala, Aug 21 2025: (Start)
Also numbers k such that (2*k - 2)!/(k!)^2 is an integer (since (2*k - 1)!/(k!)^2  + (2*k - 2)!/(k!)^2 = 2*Catalan(k-1) for k >= 1). Equivalently, numbers k such that Catalan(k-1) is divisible by k.
Since for prime p, Catalan(p-1) == -1 (mod p), the entries in this list are all nonprime. (End)

Giovanni Resta, Table of n, a(n) for n = 1..10000

Cf. A000108, A000961, A001405, A002503, A075055, A000984, A067348.

q:= k-> is(denom((2*k-1)!/(k!)^2)=1):
select(q, [$1..600])[];  # Alois P. Heinz, Feb 06 2025

Select[Range[500], IntegerQ[(2 # - 1)!/#!^2] &] (* Arkadiusz Wesolowski, Jul 01 2012 *)

Appears to be A067348(n)/2. - Benoit Cloitre, Mar 21 2003

Terms >1 are given by A002503+1. - Benoit Cloitre, Dec 09 2017

Name changed by Arkadiusz Wesolowski, Jul 01 2012

A160516 Inverse permutation to A075075.

Original entry on oeis.org

1, 2, 5, 3, 6, 4, 17, 8, 10, 7, 18, 9, 23, 20, 11, 13, 24, 15, 58, 12, 16, 19, 59, 14, 33, 22, 28, 21, 62, 26, 63, 31, 29, 25, 34, 36, 66, 57, 39, 32, 67, 35, 72, 30, 27, 60, 125, 37, 49, 44, 40, 38, 126, 47, 45, 42, 71, 61, 131, 56, 134, 64, 48, 52, 80, 46, 135, 41, 76, 43

Offset: 1

Hagen von Eitzen, May 16 2009

nonn

This is a permutation of the positive integers (provided A075075 really is a permutation).

			A075075(7) = 10, therefore a(10) = 7.
A075055(17) = 7, therefore a(7) = 17.

H. v. Eitzen, Table of n, a(n) for n = 1..50000

Cf. A185635 (fixed points).

import Data.List (elemIndex)
import Data.Maybe (fromJust)
a160516 = (+ 1) . fromJust . (`elemIndex` a075075_list)
-- Reinhard Zumkeller, Dec 19 2012

f[s_List] := Block[{m = Numerator[s[[ -1]]/s[[ -2]]]}, k = m; While[MemberQ[s, k], k += m]; Append[s, k]]; s = Nest[f, {1, 2}, 200]; Table[ Position[s, n, 1, 1], {n, 70}] // Flatten (* Robert G. Wilson v, May 20 2009 *)

A075075(a(n)) = n.

A075054 Smallest k such that (n+1)(n+2)...(n+k) is divisible by n!.

Original entry on oeis.org

1, 2, 3, 4, 5, 4, 7, 8, 9, 10, 11, 12, 13, 14, 13, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 26, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 40, 43, 44, 43, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 64, 67, 68, 69, 70, 71, 72

Offset: 1

Amarnath Murthy, Sep 07 2002

nonn

a(n) <= n. a(n) < n rarely, e.g. for n = 6, 15 etc. a(p) = p, p is a prime.

			a(6) = 4 as 7*8*9*10 is divisible by 6!= 720.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A075055, A058008.

dnf[n_]:=Module[{nf=n!,k=1},While[!Divisible[Times@@Range[ n+1,n+k],nf],k++];k]; Array[dnf,80] (* Harvey P. Dale, Jun 19 2012 *)

More terms from Sascha Kurz, Feb 02 2003

Edited by Charles R Greathouse IV, Aug 02 2010

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A058008 Numbers k such that (2*k - 1)!/(k!)^2 is an integer.

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A160516 Inverse permutation to A075075.

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A075054 Smallest k such that (n+1)(n+2)...(n+k) is divisible by n!.

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