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 1-7 of 7 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

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

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

Links

Crossrefs

Cf. A057142, A057144, A057338.

Programs

  • Haskell
    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

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

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

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 smallest of these numbers is 12, so a(7) = 12.

Links

Crossrefs

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

Programs

  • Java
    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

Extensions

More terms from David W. Wilson, Aug 28 2001

A057341 Occurrences of most frequently occurring number in 1-to-n 4-dimensional multiplication table.

Original entry on oeis.org

1, 6, 12, 28, 36, 60, 60, 96, 132, 180, 180, 276, 276, 276, 396, 468, 468, 576, 576, 756, 864, 864, 864, 1032, 1032, 1032, 1096, 1320, 1320, 1644, 1644, 1740, 1740, 1740, 2052, 2328, 2328, 2328, 2328, 2652, 2652, 3252, 3252, 3252, 3672, 3672, 3672, 4080, 4080, 4080
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. They occur 60 times, so a(7) = 60.

Crossrefs

Cf. A057142, A057338, A057342, A057343, A057344.

Extensions

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

A057344 Occurrences of most frequently occurring number in 1-to-n 5-dimensional multiplication table.

Original entry on oeis.org

1, 10, 30, 95, 140, 300, 300, 480, 740, 1260, 1260, 2070, 2070, 2640, 3780, 4620, 4620, 5810, 5810, 7530, 10660, 10660, 10660, 13840, 13840, 13840, 15340, 20340, 20340, 25020, 25020, 28320, 28320, 28320, 32710, 40090, 40090, 40090, 40090, 49740, 49740, 58980
Offset: 1

Views

Author

Neil Fernandez, Aug 28 2000

Keywords

Examples

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

Crossrefs

Cf. A057142, A057338, A057341, A057345, A057346.

Extensions

More terms from Sean A. Irvine, Jun 03 2022
Showing 1-7 of 7 results.

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