cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

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

Original entry on oeis.org

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

Views

Author

Arran Fernandez, Aug 13 2000

Keywords

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

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

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

Views

Author

Arran Fernandez, Aug 13 2000

Keywords

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(5), the most frequently occurring number is 4. It occurs 3 times, so a(5) = 3.

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).

Crossrefs

Cf. A057143, A057144, A057338.

Programs

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

Extensions

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

A057339 Largest of the most frequently occurring numbers in 1-to-n multiplication cube.

Original entry on oeis.org

1, 4, 6, 12, 20, 24, 24, 48, 72, 120, 120, 120, 120, 168, 120, 240, 240, 360, 360, 360, 360, 360, 360, 720, 720, 720, 720, 1008, 1008, 720, 720, 720, 720, 720, 1680, 2520, 2520, 2520, 2520, 1440, 1440, 2520, 2520, 2520, 2520, 2520, 2520, 5040, 5040, 5040
Offset: 1

Views

Author

Neil Fernandez, Aug 28 2000

Keywords

Examples

			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 largest of these numbers is 24, so a(7) = 24.

Links

Crossrefs

Cf. A057143, A057338, A057342, A057345.

Programs

  • Java
    public class LargestMultCube {
    static int high, 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+" "+high); } }
    private static void tally(int number, int repeatFactor) {
        counters[number] += repeatFactor;
        if(counters[number] >= highestFrequency) {
            if (counters[number] == highestFrequency)
                if (number > high) high = number;
            if (counters[number] > highestFrequency) {
                highestFrequency = counters[number]; high = number; } } } }
// Branden Aldridge, Apr 15 2022

Extensions

More terms from David W. Wilson, Aug 28 2001

A057342 Largest of the most frequently occurring numbers in 1-to-n 4-dimensional multiplication table.

Original entry on oeis.org

1, 4, 18, 24, 60, 120, 168, 96, 144, 360, 360, 720, 720, 1008, 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

Views

Author

Neil Fernandez, Aug 28 2000

Keywords

Examples

			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 largest of these numbers is 168, so a(7) = 168.

Crossrefs

Cf. A057143, A057339, A057341, A057343, A057345.

Extensions

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

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

Views

Author

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

Keywords

Comments

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

Examples

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

Links

Crossrefs

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

Programs

  • Maple
    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

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

Views

Author

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

Keywords

Examples

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

Links

Crossrefs

Cf. A064047, A057142, A057143, A057144.

Programs

  • Mathematica
    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 *)
  • PARI
    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
Showing 1-6 of 6 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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