A057144 Smallest of the most frequently occurring numbers in 1-to-n multiplication table.

1, 2, 2, 4, 4, 6, 6, 6, 6, 6, 6, 12, 12, 12, 12, 12, 12, 36, 36, 60, 60, 60, 60, 24, 24, 24, 24, 24, 24, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 120, 120, 120, 120, 120, 120, 120, 120, 120, 120, 120, 120, 120, 120, 120, 120, 120, 120, 120, 120, 120, 120, 120, 120, 120

Offset: 1

Author

Arran Fernandez, Aug 13 2000

Keywords

• nonn

Examples

			M(n) is the array in which m(x,y)= x*y for x = 1 to n and y = 1 to n. In m(10), the most frequently occurring numbers are 6, 8, 10, 12, 18, 20, 24, 30,40, each occurring 4 times. The smallest of these numbers is 6, so a(10) = 6.
		

Links

• Branden Aldridge, Table of n, a(n) for n = 1..20000 (terms 1..1000 from Reinhard Zumkeller, terms 1001..10000 from Reinhard Zumkeller and Charles R Greathouse IV)
• Benjamin Dickman, What number appears most often in an n X n multiplication table?, Mathematics StackExchange, May 2014.

Crossrefs

Cf. A057142, A057143, A057338.

Programs

Haskell

import Data.List (sort, group, sortBy, groupBy)
import Data.Function (on)
a057144 n = head $ last $ head $ groupBy ((==) `on` length) $
            reverse $ sortBy (compare `on` length) $
            group $ sort [u * v | u <- [1..n], v <- [1..n]]
-- Reinhard Zumkeller, Jun 22 2013

PARI

T(n,f=factor(n))=my(k=#f~); f[,1]=primes(k+1)[2..k+1]~; f[1,1]=6; factorback(f)
listA025487(Nmax)=vecsort(concat(vector(logint(Nmax, 2), n, select(t->t<=Nmax, if(n>1, [factorback(primes(#p), Vecrev(p))|p<-partitions(n)], [1, 2])))))
ct(n,k)=sumdiv(n,d,max(d,n/d)<=k)
a(n)=if(n==1, return(1)); my(v=listA025487(n^2),r,t,at); for(i=1,#v, t=ct(v[i],n); if(t>r, r=t; at=v[i])); at \\ Charles R Greathouse IV, Feb 05 2022

Extensions

More terms from Larry Reeves (larryr(AT)acm.org), Apr 18 2001