A065887 -id:A065887 - cp's OEIS frontend

A232098 a(n) is the largest m such that m! divides n^2; a(n) = A055881(n^2).

1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 4, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 4, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 4, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 4, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 6, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 4, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 4, 1, 2, 1

Offset: 1

Antti Karttunen, Nov 18 2013

nonn

For all n, A055881(n) <= a(n), and probably also a(n) <= A055874(n).
Moreover, a(n) > A055881(n) if and only if A055874(n) > A055881(n), thus A055926 gives (also) all the positions where this sequence differs from A055881. Please see Comments section in A055926 for the proof.
Differs from A055874 for the first time at n=840, where a(840)=7, while A055874(840)=8. A232099 gives all the positions where such differences occur.

Antti Karttunen, Table of n, a(n) for n = 1..10080

Cf. A055881, A055874, A055926, A065887, A232096, A232099, A233267.

Module[{nn=10,fct},fct=Table[{f,f!},{f,nn}];Table[Select[fct,Mod[n^2,#[[2]]]==0&][[-1,1]],{n,90}]] (* Harvey P. Dale, Aug 11 2024 *)

(define (A232098 n) (A055881 (A000290 n)))

a(n) = A055881(A000290(n)) = A055881(n^2).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{k>=1} 1/A065887(k) = 1.78672776922161809767... . - Amiram Eldar, Jan 01 2024

A065886 Smallest square divisible by n!.

Original entry on oeis.org

1, 1, 4, 36, 144, 3600, 3600, 176400, 2822400, 25401600, 25401600, 3073593600, 110649369600, 18699743462400, 74798973849600, 1869974346240000, 29919589539840000, 8646761377013760000, 77820852393123840000, 28093327713917706240000, 112373310855670824960000

Offset: 0

Henry Bottomley, Nov 27 2001

nonn

			a(10) = 25401600 since 10! = 3628800 and the smallest square divisible by this is 25401600 = 3628800*7 = 5040^2

Robert Israel, Table of n, a(n) for n = 0..407

N:= 50: # to get a(0)..a(N)
P:= select(isprime, [$2..N]):
nP:= nops(P):
V:= Vector(nP):
A[0]:= 1:
for n from 1 to N do
  for i from 1 to nP do V[i]:= V[i] + padic:-ordp(n,P[i]) od;
  A[n]:= mul(P[i]^(2*ceil(V[i]/2)),i=1..nP)
od:
seq(A[n],n=0..N); # Robert Israel, Jan 30 2017

ssd[n_]:=Module[{nf=n!,k=1},While[!IntegerQ[Sqrt[k*nf]],k++];k*nf]; Array[ssd,20,0] (* Harvey P. Dale, Apr 29 2012 *)

a(n) = A053143(A000142(n)) = A065887(n)^2 = A000142(n)*A055204(n) = A001044(n)/A055071(n)

cp's OEIS Frontend

A232098 a(n) is the largest m such that m! divides n^2; a(n) = A055881(n^2).

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