A057144 -id:A057144 - cp's OEIS frontend

A333665 Numbers that occur in A057144 arranged in ascending order.

1, 2, 4, 6, 12, 24, 36, 60, 120, 240, 360, 720, 840, 1680, 2520, 5040, 10080, 27720, 30240, 55440, 110880, 166320, 332640, 720720, 1441440, 2162160, 3603600, 4324320

Offset: 1

Dmitry Kamenetsky, Sep 03 2020

30240 is the smallest term that is not highly composite (A002182).
From Hal M. Switkay, Jul 25 2022: (Start)
It appears that each term is the least number having its prime signature (A025487).
Although it would be desirable to produce more terms of this sequence, the first 28 terms represent the union of at least the first 20000 terms of A057144.
The smallest of the most frequently occurring numbers in the 1-to-k multiplication table, A057144(k), do not constitute a nondecreasing function of k. Thus the minimal values of k corresponding to terms of this sequence do not form an increasing sequence. For example, A057144(k) = 60 when 20 <= k <= 23 or 30 <= k <= 39; but A057144(k) = 24 when 24 <= k <= 29, and A057144(k) = 36 when 18 <= k <= 19. (End)

			For 6 <= k <= 11, the smallest of the most frequently occurring numbers in the 1-to-k multiplication table is 6. a(4) = 6, because it is the fourth natural number to be the smallest of the most frequently occurring numbers in the 1-to-k multiplication table. - _Hal M. Switkay_, Jul 27 2022

Benjamin Dickman, What number appears most often in a n X n multiplication table?, Mathematics StackExchange, May 2014.

Cf. A002182, A057144.

a(20)-a(28) from Hal M. Switkay, using the b-file of A057144, Jul 25 2022

A057142 Occurrences of most frequently occurring number in 1-to-n multiplication table.

Original entry on oeis.org

1, 2, 2, 3, 3, 4, 4, 4, 4, 4, 4, 6, 6, 6, 6, 6, 6, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 16, 16, 16, 16, 16, 16, 16, 16

Offset: 1

Arran Fernandez, Aug 13 2000

nonn

			M(n) is the array in which m(x,y)= x*y for x = 1 to n and y = 1 to n. In m(5), the most frequently occurring number is 4. It occurs 3 times, so a(5) = 3.

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).

Cf. A057143, A057144, A057338.

import Data.List (group, sort)
a057142 n = head $ reverse $ sort $ map length $ group $
            sort [u * v | u <- [1..n], v <- [1..n]]
-- Reinhard Zumkeller, Jun 22 2013

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); for(i=1,#v, t=ct(v[i],n); if(t>r, r=t)); r \\ Charles R Greathouse IV, Feb 05 2022

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

A057143 Largest of the most frequently occurring numbers in 1-to-n multiplication table.

Original entry on oeis.org

1, 2, 6, 4, 4, 12, 12, 24, 24, 40, 40, 24, 24, 24, 60, 60, 60, 36, 36, 60, 60, 60, 60, 120, 120, 120, 120, 168, 168, 120, 120, 120, 120, 120, 120, 180, 180, 180, 180, 120, 120, 120, 120, 120, 360, 360, 360, 360, 360, 360, 360, 360, 360, 360, 360, 360, 360, 360

Offset: 1

Arran Fernandez, Aug 13 2000

nonn

			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 largest of these numbers is 40, so a(10) = 40.

Branden Aldridge, Table of n, a(n) for n = 1..20000 (first 1000 terms from Reinhard Zumkeller).

Cf. A057142, A057144, A057338.

import Data.List (group, sort, sortBy)
import Data.Function (on)
a057143 n = head $ head $ reverse $ sortBy (compare `on` length) $
            group $ sort [u * v | u <- [1..n], v <- [1..n]]
-- Reinhard Zumkeller, Jun 22 2013

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

A057340 Smallest of the most frequently occurring numbers in 1-to-n multiplication cube.

Original entry on oeis.org

1, 2, 6, 12, 12, 12, 12, 24, 72, 60, 60, 72, 72, 72, 120, 240, 240, 180, 180, 360, 360, 360, 360, 720, 720, 720, 720, 720, 720, 720, 720, 720, 720, 720, 840, 2520, 2520, 2520, 2520, 1440, 1440, 2520, 2520, 2520, 2520, 2520, 2520, 5040, 5040, 5040, 5040

Offset: 1

Neil Fernandez, Aug 28 2000

nonn

			M(n) is the array in which m(x,y,z)=x*y*z for x = 1 to n, y = 1 to n and z = 1 to n. In M(7), the most frequently occurring numbers are 12 and 24, each occurring 15 times. The smallest of these numbers is 12, so a(7) = 12.

Branden Aldridge and David A. Corneth, Table of n, a(n) for n = 1..10000 (first 500 terms from Branden Aldridge)

Cf. A025487, A057144, A057338, A057343, A057346.

public class SmallestMultCube {
    static int low, highestFrequency = 0;
    static int[] counters;
    public static void main(String[] args) {
        int max=500;
        counters = new int[max*max*max+1];
        for(int outer=1; outer<=max; outer++) {
            tally(outer*outer*outer, 1);
            for(int middle=outer-1; middle>=1; middle--) {
                tally(outer*outer*middle, 3); tally(outer*middle*middle, 3);
                for(int inner=middle-1; inner>=1; inner--) {
                    tally(outer*middle*inner, 6); } }
            System.out.println(outer+" "+low); } }
    private static void tally(int number, int repeatFactor) {
        counters[number] += repeatFactor;
        if(counters[number] >= highestFrequency) {
            if (counters[number] == highestFrequency)
                if (number < low) low = number;
            if (counters[number] > highestFrequency) {
                highestFrequency = counters[number]; low = number; } } } }
// Branden Aldridge, Apr 15 2022

More terms from David W. Wilson, Aug 28 2001

A057343 Smallest of the most frequently occurring numbers in 1-to-n 4-dimensional multiplication table.

Original entry on oeis.org

1, 4, 6, 24, 60, 60, 60, 96, 144, 360, 360, 720, 720, 720, 720, 1440, 1440, 1440, 1440, 1440, 5040, 5040, 5040, 5040, 5040, 5040, 4320, 10080, 10080, 10080, 10080, 10080, 10080, 10080, 10080, 30240, 30240, 30240, 30240, 30240, 30240, 30240, 30240, 30240, 30240, 30240, 30240, 30240, 30240, 30240

Offset: 1

Neil Fernandez, Aug 28 2000

nonn

			M(n) is the array in which m(x,y,z,t)=x*y*z*t for x = 1 to n, y = 1 to n, z = 1 to n and t = 1 to n In M(7), the most frequently occurring numbers are 60, 72, 84, 120 and 168, each occurring 60 times. The smallest of these numbers is 60, so a(7) = 60.

Cf. A057144, A057340, A057341, A057342, A057346.

a(16)-a(50) from Charlie Neder, Jan 15 2019

A057346 Smallest of the most frequently occurring numbers in 1-to-n 5-dimensional multiplication table.

Original entry on oeis.org

1, 4, 12, 48, 120, 360, 360, 480, 864, 1440, 1440, 1440, 1440, 5040, 5040, 10080, 10080, 8640, 8640, 8640, 30240, 30240, 30240, 60480, 60480, 60480, 60480, 60480, 60480, 60480, 60480, 120960, 120960, 120960, 302400, 302400, 302400, 302400, 302400, 302400, 302400, 302400

Offset: 1

Neil Fernandez, Aug 28 2000

nonn

			M(n) is the array in which m(x,y,z,t,u)=x*y*z*t*u for x = 1 to n, y = 1 to n, z = 1 to n, t = 1 to n and u = 1 to n. In M(7), the most frequently occurring numbers are 360, 420, 504 and 840, each occurring 300 times. The smallest of these numbers is 360, so a(7) = 360.

David A. Corneth, Table of n, a(n) for n = 1..100

Cf. A057144, A057340, A057343, A057344.

More terms from David A. Corneth, Apr 18 2022

A064047 Number of numbers only appearing once in 1-to-n multiplication table.

Original entry on oeis.org

1, 2, 3, 3, 4, 5, 6, 6, 5, 6, 7, 8, 9, 10, 11, 10, 11, 12, 13, 13, 14, 15, 16, 17, 15, 16, 15, 15, 16, 17, 18, 17, 18, 19, 20, 20, 21, 22, 23, 24, 25, 26, 27, 27, 28, 29, 30, 30, 26, 26, 27, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 36, 37, 38, 39, 39, 40, 41, 42, 42, 43

Offset: 1

Matthew Somerville (matthew.somerville(AT)trinity.oxford.ac.uk), Aug 24 2001

nonn

For n <= 127, this is the same as the number of vertices of the polytope representing the number n. The latter is given in A335152. The sequences differ starting at n = 128. See A335152 and Lu and Deng, Appendix. - N. J. A. Sloane, May 25 2020

a(n) is the number of x in [1,n] such that x^2 has no divisor d with x < d <= n. - Robert Israel, Sep 03 2020

			In the 1-to-5 multiplication table, four numbers (1,9,16,25) appear once only. Therefore a(5)=4.

Robert Israel, Table of n, a(n) for n = 1..10000
Ya-Ping Lu and Shu-Fang Deng, Properties of Polytopes Representing Natural Numbers, arXiv:2003.08968 [math.GM], 2020.

Cf. A064048, A057142, A057143, A057144, A335152.

N:= 200: # for a(1)..a(N)
V:= Vector(N):
for x from 1 to N do
  y:= min(N, min(select(`>`,numtheory:-divisors(x^2),x))-1);
  V[x..y]:= map(`+`,V[x..y],1)
od:
convert(V,list); # Robert Israel, Sep 03 2020

A057345 Largest of the most frequently occurring numbers in 1-to-n 5-dimensional multiplication table.

Original entry on oeis.org

1, 8, 36, 48, 120, 360, 840, 672, 864, 1440, 1440, 1440, 1440, 5040, 5040, 10080, 10080, 8640, 8640, 8640, 30240, 30240, 30240, 60480, 60480, 60480, 60480, 60480, 60480, 60480, 60480, 120960, 120960, 120960, 302400, 302400, 302400, 302400, 302400, 302400

Offset: 1

Neil Fernandez, Aug 28 2000

nonn

			M(n) is the array in which m(x,y,z,t,u)=x*y*z*t*u for x = 1 to n, y = 1 to n, z = 1 to n, t = 1 to n and u = 1 to n. In M(7), the most frequently occurring numbers are 360, 420, 504 and 840, each occurring 300 times. The largest of these numbers is 840, so a(7) = 840.

Cf. A057144, A057339, A057342, A057344.

More terms from Sean A. Irvine, Jun 03 2022

A064048 Number of most frequently occurring numbers in the 1-to-n multiplication table.

Original entry on oeis.org

1, 1, 3, 1, 1, 2, 2, 4, 5, 9, 9, 2, 2, 2, 4, 5, 5, 1, 1, 1, 1, 1, 1, 5, 5, 5, 5, 7, 7, 2, 2, 2, 2, 2, 2, 4, 4, 4, 4, 1, 1, 1, 1, 1, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 6, 6, 6, 6, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 6, 6, 1, 1, 1, 1, 1, 1, 1, 1, 3, 3, 3, 3, 6, 6, 6, 6, 6, 6, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2

Offset: 1

Matthew Somerville (matthew.somerville(AT)trinity.oxford.ac.uk), Aug 24 2001

nonn

			In the 1-to-6 multiplication table, the most frequently occurring numbers (each occurring 4 times) are 6 and 12. Therefore a(6)=2.

Branden Aldridge, Table of n, a(n) for n = 1..20000

Cf. A064047, A057142, A057143, A057144.

mfon[n_]:=Module[{x=SortBy[Tally[Times@@@Tuples[Range[n],2]], Last]}, Length[ Select[x,#[[2]]==x[[-1,2]]&]]]; Array[mfon,100] (* Harvey P. Dale, Oct 20 2012 *)

a(n)=my(v=List(),ct,s,r,t); for(a=1,n,for(b=1,n,listput(v,a*b))); s=Set(v); for(i=1,#s, t=sum(j=1,#v,v[j]==s[i]); if(tr,ct=1;r=t, ct++)); ct \\ Charles R Greathouse IV, Feb 05 2022

cp's OEIS Frontend

A333665 Numbers that occur in A057144 arranged in ascending order.

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A057142 Occurrences of most frequently occurring number in 1-to-n multiplication table.

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A057143 Largest of the most frequently occurring numbers in 1-to-n multiplication table.

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A057340 Smallest of the most frequently occurring numbers in 1-to-n multiplication cube.

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Java

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A057343 Smallest of the most frequently occurring numbers in 1-to-n 4-dimensional multiplication table.

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A057346 Smallest of the most frequently occurring numbers in 1-to-n 5-dimensional multiplication table.

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A064047 Number of numbers only appearing once in 1-to-n multiplication table.

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Maple

A057345 Largest of the most frequently occurring numbers in 1-to-n 5-dimensional multiplication table.

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A064048 Number of most frequently occurring numbers in the 1-to-n multiplication table.

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