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

1,2

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.

Example

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.

Prog

(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

Crossrefs

Cf. A057142, A057143, A057338.

Keyword

nonn

Author

_Arran Fernandez_, Aug 13 2000

Extensions

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

Status

approved

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

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

1,2

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).

Example

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.

Prog

(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

Crossrefs

Cf. A057143, A057144, A057338.

Keyword

nonn

Author

_Arran Fernandez_, Aug 13 2000

Extensions

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

Status

approved

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

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

1,2

Links

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

Example

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.

Prog

(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

Crossrefs

Cf. A057142, A057144, A057338.

Keyword

nonn

Author

_Arran Fernandez_, Aug 13 2000

Extensions

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

Status

approved

A145193 Omega(6n-1) + Omega(6n+1).

2, 2, 2, 3, 2, 3, 2, 3, 3, 2, 3, 2, 3, 3, 3, 3, 2, 2, 3, 4, 4, 3, 2, 4, 2, 3, 3, 3, 4, 2, 4, 2, 2, 4, 3, 4, 3, 2, 3, 2, 5, 3, 3, 3, 2, 4, 2, 4, 3, 4, 3, 2, 3, 5, 3, 3, 5, 2, 3, 3, 3, 3, 3, 4, 3, 3, 3, 3, 4, 2, 5, 2, 3, 3, 3, 4, 2, 3, 5, 3, 3, 3, 3, 3, 3

Offset

1,1

Links

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Formula

a(n) = A001222(6n-1) + A001222(6n+1). - Michel Marcus, Sep 04 2013

Example

For n=1, Omega(6n-1) + Omega(6n+1) = Omega(5) + Omega(7) = 1+1 = 2, so a(1)=2.

Mathematica

For[x=6, x<601, x+=6, S=0; T=0; For[k=1, k< Length[FactorInteger[x-1]]+1, k++, S+= FactorInteger[x-1][[k]][[2]]]; For[m=1, m<Length[FactorInteger[x+1]]+1, m++, T+= FactorInteger[x+1][[m]][[2]]]; Print[x/6, " ", S+T]]
Table[Total[PrimeOmega/@(6n+{1, -1})], {n, 60}] (* Harvey P. Dale, May 22 2013 *)

Prog

(PARI) a(n) = bigomega(6*n-1) + bigomega(6*n+1); \\ Michel Marcus, Sep 04 2013

Crossrefs

Cf. A145194.

Keyword

nonn,easy

Author

_Arran Fernandez_, Oct 03 2008

Extensions

More terms from Michel Marcus, Sep 04 2013

Status

approved

A078467 a(n) = a(n-1) + a(n-4); first four terms are 0,1,2,3.

0, 1, 2, 3, 3, 4, 6, 9, 12, 16, 22, 31, 43, 59, 81, 112, 155, 214, 295, 407, 562, 776, 1071, 1478, 2040, 2816, 3887, 5365, 7405, 10221, 14108, 19473, 26878, 37099, 51207, 70680, 97558, 134657, 185864, 256544, 354102, 488759, 674623, 931167, 1285269

Offset

0,3

Links

Table of n, a(n) for n=0..44.

Index entries for linear recurrences with constant coefficients, signature (1, 0, 0, 1).

Formula

a(n) = a(n-1) + a(n-4); a(0)=0, a(1)=1, a(2)=2, a(3)=3

a(n+1)=sum{k=0..n, binomial(n-k, floor(k/3))} - Paul Barry, Jul 06 2004

G.f.: x(1+x+x^2)/(1-x-x^4). a(n)=A003269(n)+A003269(n-1)+A003269(n-2). [From R. J. Mathar, Nov 25 2008]

Example

The sequence begins 0,1,2,3. a(5) = a(5-1) + a(5-4) = a(4)+a(1)= 3+0 =3. a(6) = a(6-1) + a(6-4) = a(5) + a(2) = 3+1 = 4.

Mathematica

LinearRecurrence[{1, 0, 0, 1}, {0, 1, 2, 3}, 50] (* Harvey P. Dale, Oct 08 2012 *)

Crossrefs

Cf. A000045, A003269, A058278.

Keyword

nonn

Author

_Arran Fernandez_, Jan 02 2003

Status

approved

A145192 Integers n for which Omega(6n-1)>2 and Omega(6n+1)>2

141, 421, 479, 596, 629, 746, 801, 804, 904, 966, 981, 1016, 1042, 1051, 1119, 1121, 1142, 1146, 1154, 1261, 1289, 1296, 1324, 1329, 1384, 1399, 1406, 1454, 1471, 1493, 1499, 1560, 1576, 1597, 1637, 1646

Offset

1,1

Links

Harvey P. Dale, Table of n, a(n) for n = 1..10000

Example

(6*141)-1 = 845, which has >2 prime factors (counted with multiplicity), namely 5,13 and 13. (6*141)+1 = 847, which has >2 prime factors (counted with multiplicity), namely 7,11 and 11. So 141 is in the sequence.

Mathematica

For[x = 6, x < 10001, x += 6, If[PrimeQ[x - 1] == True, y = "P", S = 0; F = FactorInteger[x - 1]; For[k = 1, k < Length[F] + 1, k++, S += F[[k]][[2]]]; If[S == 2, y = "A", y = "N"]]; If[PrimeQ[x + 1] == True, z = "P", S = 0; F = FactorInteger[x + 1]; For[k = 1, k < Length[F] + 1, k++, S += F[[k]][[2]]]; If[S == 2, z = "A", z = "N"]]; If[y == "N" && z == "N", Print[x/6]]]
Select[Range[2000], PrimeOmega[6#+1]>2&&PrimeOmega[6#-1]>2&] (* Harvey P. Dale, Apr 26 2016 *)

Crossrefs

Cf. A001222.

Keyword

nonn

Author

_Arran Fernandez_, Oct 03 2008

Status

approved

A145194 Least k for which Omega(6k-1) + Omega(6k+1) >= n.

1, 1, 4, 20, 41, 104, 479, 1146, 7603, 16521, 91146, 188021, 188021, 1861979, 14122396, 43294271, 203450521, 203450521, 5877278646, 5900065104, 16886393229

Offset

1,3

Comments

a(22) <= 170499674479. a(23) <= 1307169596354. a(24) <= 3178914388021. a(25) <= 3178914388021. a(26) <= 43614705403646 [From Donovan Johnson, Feb 17 2010]

Links

Table of n, a(n) for n=1..21.

Example

When k=1,2 and 3, Omega(6k-1) + Omega(6k+1) = 2. When k=4, Omega(6k-1) + Omega(6k+1) = 3, so a(3)=4.

Mathematica

Maxie=0; For[x=6, x<10000001, x+=6, S=0; T=0; For[k=1, k< Length[FactorInteger[x-1]]+1, k++, S+= FactorInteger[x-1][[k]][[2]]]; For[m=1, m< Length[FactorInteger[x+1]]+1, m++, T+= FactorInteger[x+1][[m]][[2]]]; If[S+T>Maxie, Print[x/6, " ", S+T]; Maxie=S+T]]

Crossrefs

Cf. A145193

Keyword

nonn

Author

_Arran Fernandez_, Oct 03 2008

Extensions

a(15)-a(21) from Donovan Johnson, Feb 17 2010

Status

approved

A225778 Continued fraction expansion of the smallest positive solution of Gamma(-x)=Gamma(-2x)

0, 2, 1, 50, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 3, 2, 1, 34, 1, 4, 1, 1, 6, 2, 3, 6, 3, 43, 1, 1, 1, 5, 1, 1, 2, 2, 1, 1, 1, 3, 1, 15, 3, 2, 3, 5, 5, 1, 8, 1, 5, 1, 3, 2, 1, 2, 1, 1, 1, 1, 42, 3, 6, 12, 1, 2, 4, 8, 1, 7, 2, 1, 2, 3, 1, 154, 1, 1, 1, 1, 11, 1, 4, 3, 1, 8, 5, 2, 3, 2

Offset

0,2

Links

Table of n, a(n) for n=0..90.

Example

0.335514419... = 0+1/(2+1/(1+...))

Mathematica

ContinuedFraction[FindRoot[Gamma[-x] - Gamma[-2x] == 0, {x, 0.3}, WorkingPrecision -> 100]]

Prog

(PARI) contfrac(solve(x=.3, .4, gamma(-x)-gamma(-2*x))) \\ Charles R Greathouse IV, Jul 26 2013

Keyword

nonn,cofr

Author

_Arran Fernandez_, Jul 26 2013

Status

approved