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.

User: Arran Fernandez

Arran Fernandez's wiki page.

Arran Fernandez has authored 8 sequences.

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

Original entry on oeis.org

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

Views

Author

Arran Fernandez, Jul 26 2013

Keywords

Examples

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

Programs

  • Mathematica
    ContinuedFraction[FindRoot[Gamma[-x] - Gamma[-2x] == 0, {x, 0.3}, WorkingPrecision -> 100]]
  • PARI
    contfrac(solve(x=.3,.4,gamma(-x)-gamma(-2*x))) \\ Charles R Greathouse IV, Jul 26 2013

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

Original entry on oeis.org

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

Views

Author

Arran Fernandez, Oct 03 2008

Keywords

Comments

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

Examples

			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.

Crossrefs

Cf. A145193

Programs

  • 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]]

Extensions

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

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

Original entry on oeis.org

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

Views

Author

Arran Fernandez, Oct 03 2008

Keywords

Examples

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

Links

Crossrefs

Cf. A145194.

Programs

  • 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, mHarvey P. Dale, May 22 2013 *)
  • PARI
    a(n) = bigomega(6*n-1) + bigomega(6*n+1); \\ Michel Marcus, Sep 04 2013

Formula

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

Extensions

More terms from Michel Marcus, Sep 04 2013

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

Original entry on oeis.org

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

Views

Author

Arran Fernandez, Oct 03 2008

Keywords

Examples

			(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.

Links

Crossrefs

Cf. A001222.

Programs

  • 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 *)

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

Original entry on oeis.org

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

Views

Author

Arran Fernandez, Jan 02 2003

Keywords

Examples

			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.

Links

Crossrefs

Cf. A000045, A003269, A058278.

Programs

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

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]

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

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

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

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