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.

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A079209 Number of isomorphism classes of associative commutative non-anti-associative non-anti-commutative closed binary operations on a set of order n, listed by class size.

Original entry on oeis.org

0, 0, 3, 0, 0, 3, 9, 0, 0, 0, 3, 0, 0, 16, 39
Offset: 1

Views

Author

Christian van den Bosch (cjb(AT)cjb.ie), Jan 03 2003

Keywords

Comments

A079202(n)+A079203(n)+A079204(n)+A079205(n)+A079197(n)+A079207(n)+A079208(n)+A079209(n)+A063524(n)=A079171(n)
Elements per row: 1,2,4,8,16,30,... (given by A027423, number of positive divisors of n!)
First four rows: 0; 0,3; 0,0,3,9; 0,0,0,3,0,0,16,39
A079244(x) is equal to the sum of the products of each element in row x of this sequence and the corresponding element of A079210.
The sum of each row x of this sequence is given by A079245(x).

Links

Crossrefs

Cf. A079202, A079203, A079204, A079205, A079197, A079207, A079208, A079244, A079245.

A336941 Number of strict chains of divisors starting with the superprimorial A006939(n) and ending with 1.

Original entry on oeis.org

1, 1, 8, 604, 691968, 16359233536, 10083474928244288, 195661337707783118840768, 139988400203593571474134024847360, 4231553868972506381329450624389969130848256, 6090860257621637852755610879241895108657182173073604608, 464479854191019594417264488167571483344961210693790188774166838214656
Offset: 0

Views

Author

Gus Wiseman, Aug 13 2020

Keywords

Examples

			The a(2) = 8 chains:
  12/1
  12/2/1
  12/3/1
  12/4/1
  12/6/1
  12/4/2/1
  12/6/2/1
  12/6/3/1

Links

Crossrefs

A022915 is the maximal case.
A076954 can be used instead of A006939.
A336571 is the case with distinct prime multiplicities.
A336942 is the case using members of A130091.
A337070 is the version ending with any divisor of A006939(n).
A000005 counts divisors.
A074206 counts chains of divisors from n to 1.
A006939 lists superprimorials or Chernoff numbers.
A067824 counts divisor chains starting with n.
A181818 gives products of superprimorials, with complement A336426.
A253249 counts chains of divisors.
A317829 counts factorizations of superprimorials.
A336423 counts chains using A130091, with maximal case A336569.
Cf. A000142, A001055, A002033, A008480, A022559, A027423, A124010, A167865, A181796, A336417, A336420, A337069.

Programs

  • Mathematica
    chern[n_]:=Product[Prime[i]^(n-i+1),{i,n}];
chns[n_]:=If[n==1,1,Sum[chns[d],{d,Most[Divisors[n]]}]];
Table[chns[chern[n]],{n,0,3}]
  • PARI
    a(n)={my(sig=vector(n,i,i), m=vecsum(sig)); sum(k=0, m, prod(i=1, #sig, binomial(sig[i]+k-1, k-1))*sum(r=k, m, binomial(r,k)*(-1)^(r-k)))} \\ Andrew Howroyd, Aug 30 2020

Formula

a(n) = A337070(n)/2 for n > 0.
a(n) = A074206(A006939(n)).

Extensions

Terms a(8) and beyond from Andrew Howroyd, Aug 30 2020

A064145 a(n) = tau(n!-1) or number of divisors of n!-1.

Original entry on oeis.org

1, 2, 2, 4, 2, 2, 4, 6, 4, 16, 2, 4, 2, 24, 4, 8, 8, 8, 4, 16, 8, 4, 4, 8, 4, 4, 16, 32, 2, 8, 2, 2, 4, 8, 4, 32, 2, 16, 4, 16, 16, 128, 16, 32, 32, 4, 16, 8, 4, 32, 32, 16, 64, 64, 32, 64, 32, 4, 8, 16, 16, 32, 16, 64, 16, 128, 4, 64, 32, 32, 8, 16, 32, 128, 8
Offset: 2

Views

Author

Vladeta Jovovic, Sep 11 2001

Keywords

Links

Crossrefs

Cf. A000005, A002582, A027423, A002981, A002982, A064144.

Programs

  • Mathematica
    Do[ Print[ DivisorSigma[0, n! - 1]], {n, 2, 40} ]
DivisorSigma[0,Range[2,80]!-1] (* Harvey P. Dale, Aug 17 2024 *)
  • PARI
    { f=1; for (n=2, 100, f*=n; if (n>1, a=numdiv(f - 1), a=0); write("b064145.txt", n, " ", a) ) } \\ Harry J. Smith, Sep 09 2009

Extensions

More terms from Robert G. Wilson v, Oct 04 2001
a(51)-a(76) from Harry J. Smith, Sep 09 2009
Ambiguous term a(1) removed by Max Alekseyev, May 06 2022

A079201 Number of isomorphism classes of associative commutative closed binary operations on a set of order n, listed by class size.

Original entry on oeis.org

1, 1, 0, 3, 0, 0, 3, 9, 0, 0, 0, 3, 0, 0, 16, 39, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 15, 0, 4, 0, 103, 201, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 12, 0, 0, 0, 0, 6, 0, 0, 4, 91, 0, 55, 0, 715, 1258, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 12
Offset: 0

Views

Author

Christian van den Bosch (cjb(AT)cjb.ie), Jan 03 2003

Keywords

Comments

Number of elements per row: 1,1,2,4,8,16,30,... (given by A027423, number of positive divisors of n!).

Examples

			Triangle T(n,k) begins:
  1;
  1;
  0, 3;
  0, 0, 3, 9;
  0, 0, 0, 3, 0, 0, 16, 39;
  0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 15, 0, 4, 0, 103, 201;

Links

Crossrefs

Row sums are A001426.
Cf. A023815, A027423 (row lengths), A079171, A079194, A079197, A079200, A058105.

Formula

A079194(n,k) + A079197(n,k) + A079200(n,k) + T(n,k) = A079171(n,k).
T(n, A027423(n)) = A058105(n).
A023815(n) = Sum_{k>=1} T(n,k)*A079210(n,k).

Extensions

a(0)=1 prepended and terms a(16) and beyond from Andrew Howroyd, Jan 26 2022

A336496 Products of superfactorials (A000178).

Original entry on oeis.org

1, 2, 4, 8, 12, 16, 24, 32, 48, 64, 96, 128, 144, 192, 256, 288, 384, 512, 576, 768, 1024, 1152, 1536, 1728, 2048, 2304, 3072, 3456, 4096, 4608, 6144, 6912, 8192, 9216, 12288, 13824, 16384, 18432, 20736, 24576, 27648, 32768, 34560, 36864, 41472, 49152, 55296
Offset: 1

Views

Author

Gus Wiseman, Aug 03 2020

Keywords

Comments

First differs from A317804 in having 34560, which is the first term with more than two distinct prime factors.

Examples

			The sequence of terms together with their prime indices begins:
    1: {}
    2: {1}
    4: {1,1}
    8: {1,1,1}
   12: {1,1,2}
   16: {1,1,1,1}
   24: {1,1,1,2}
   32: {1,1,1,1,1}
   48: {1,1,1,1,2}
   64: {1,1,1,1,1,1}
   96: {1,1,1,1,1,2}
  128: {1,1,1,1,1,1,1}
  144: {1,1,1,1,2,2}
  192: {1,1,1,1,1,1,2}
  256: {1,1,1,1,1,1,1,1}
  288: {1,1,1,1,1,2,2}
  384: {1,1,1,1,1,1,1,2}
  512: {1,1,1,1,1,1,1,1,1}

Crossrefs

A001013 is the version for factorials, with complement A093373.
A181818 is the version for superprimorials, with complement A336426.
A336497 is the complement.
A000178 lists superfactorials.
A001055 counts factorizations.
A006939 lists superprimorials or Chernoff numbers.
A049711 is the minimum prime multiplicity in A000178.
A174605 is the maximum prime multiplicity in A000178.
A303279 counts prime factors of superfactorials.
A317829 counts factorizations of superprimorials.
A322583 counts factorizations into factorials.
A325509 counts factorizations of factorials into factorials.
Cf. A000142, A000720, A007489, A011371, A022559, A022915, A027423, A034878, A034876, A076954, A115627, A294068.

Programs

  • Mathematica
    supfac[n_]:=Product[k!,{k,n}];
facsusing[s_,n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facsusing[Select[s,Divisible[n/d,#]&],n/d],Min@@#>=d&]],{d,Select[s,Divisible[n,#]&]}]];
Select[Range[1000],facsusing[Rest[Array[supfac,30]],#]!={}&]

A337074 Number of strict chains of divisors in A130091 (numbers with distinct prime multiplicities), starting with n!.

Original entry on oeis.org

1, 1, 2, 0, 28, 0, 768, 0, 0, 0, 42155360, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
Offset: 0

Views

Author

Gus Wiseman, Aug 16 2020

Keywords

Comments

Support appears to be {0, 1, 2, 4, 6, 10}.

Examples

			The a(4) = 28 chains:
  24  24/1   24/2/1   24/4/2/1   24/8/4/2/1
      24/2   24/3/1   24/8/2/1   24/12/4/2/1
      24/3   24/4/1   24/8/4/1
      24/4   24/4/2   24/8/4/2
      24/8   24/8/1   24/12/2/1
      24/12  24/8/2   24/12/3/1
             24/8/4   24/12/4/1
             24/12/1  24/12/4/2
             24/12/2
             24/12/3
             24/12/4

Crossrefs

A336867 is the complement of the support.
A336868 is the characteristic function (image under A057427).
A336942 is half the version for superprimorials (n > 1).
A337071 does not require distinct prime multiplicities.
A337104 is the case of chains ending with 1.
A000005 counts divisors.
A000142 lists factorial numbers.
A027423 counts divisors of factorial numbers.
A067824 counts chains of divisors starting with n.
A074206 counts chains of divisors from n to 1.
A076716 counts factorizations of factorial numbers.
A130091 lists numbers with distinct prime multiplicities.
A181796 counts divisors with distinct prime multiplicities.
A253249 counts chains of divisors.
A327498 gives the maximum divisor with distinct prime multiplicities.
A336414 counts divisors of n! with distinct prime multiplicities.
A336415 counts divisors of n! with equal prime multiplicities.
A336423 counts chains using A130091, with maximal case A336569.
A336571 counts chains of divisors 1 < d < n using A130091.
Cf. A000110, A001055, A002033, A007489, A022559, A048656, A071626, A098859, A124010, A167865, A325617, A336416, A336424.

Programs

  • Mathematica
    chnsc[n_]:=If[!UnsameQ@@Last/@FactorInteger[n],{},If[n==1,{{1}},Prepend[Join@@Table[Prepend[#,n]&/@chnsc[d],{d,Most[Divisors[n]]}],{n}]]];
Table[Length[chnsc[n!]],{n,0,6}]

Formula

a(n) = 2*A337104(n) = 2*A336423(n!) for n > 1.

A337107 Irregular triangle read by rows where T(n,k) is the number of strict length-k chains of divisors from n! to 1.

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 0, 1, 6, 9, 4, 0, 1, 14, 45, 52, 20, 0, 1, 28, 183, 496, 655, 420, 105, 0, 1, 58, 633, 2716, 5755, 6450, 3675, 840, 0, 1, 94, 1659, 11996, 46235, 106806, 155869, 145384, 84276, 27720, 3960
Offset: 1

Views

Author

Gus Wiseman, Aug 23 2020

Keywords

Comments

Row n > 1 appears to be row n! of A334996.

Examples

			Triangle begins:
    1
    0    1
    0    1    2
    0    1    6    9    4
    0    1   14   45   52   20
    0    1   28  183  496  655  420  105
    0    1   58  633 2716 5755 6450 3675  840
Row n = 4 counts the following chains:
  24/1  24/2/1   24/4/2/1   24/8/4/2/1
        24/3/1   24/6/2/1   24/12/4/2/1
        24/4/1   24/6/3/1   24/12/6/2/1
        24/6/1   24/8/2/1   24/12/6/3/1
        24/8/1   24/8/4/1
        24/12/1  24/12/2/1
                 24/12/3/1
                 24/12/4/1
                 24/12/6/1

Crossrefs

A097805 is the restriction to powers of 2.
A325617 is the maximal case.
A337105 gives row sums.
A337106 is column k = 3.
A000005 counts divisors.
A000142 lists factorial numbers.
A001055 counts factorizations.
A074206 counts chains of divisors from n to 1.
A027423 counts divisors of factorial numbers.
A067824 counts chains of divisors starting with n.
A076716 counts factorizations of factorial numbers.
A253249 counts chains of divisors.
A337071 counts chains starting with n!.
Cf. A022559, A124010, A167865, A251683, A337070, A337074.

Programs

  • Maple
    b:= proc(n) option remember; expand(x*(`if`(n=1, 1, 0) +
      add(b(d), d=numtheory[divisors](n) minus {n})))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..degree(p)))(b(n!)):
seq(T(n), n=1..10);  # Alois P. Heinz, Aug 23 2020
  • Mathematica
    nv=5;
chnsc[n_]:=Select[Prepend[Join@@Table[Prepend[#,n]&/@chnsc[d],{d,DeleteCases[Divisors[n],n]}],{n}],MemberQ[#,1]&];
Table[Length[Select[chnsc[n!],Length[#]==k&]],{n,nv},{k,1+PrimeOmega[n!]}]

A060742 Number of divisors of n! which are also differences between consecutive divisors of n! (ordered by size).

Original entry on oeis.org

0, 0, 1, 2, 4, 9, 15, 27, 41, 68, 111, 218, 328, 624, 929, 1518, 2016, 3689, 4965, 9252, 13177, 20016, 30697, 56749, 69434, 94242, 149558, 190292, 258370, 492924, 615063, 1149403, 1325124, 1841343, 2737190, 3592273, 4193855, 8216492, 12668800, 17654339, 20368544
Offset: 0

Views

Author

Labos Elemer, Apr 23 2001

Keywords

Examples

			For n = 5, n! = 120; divisors = {1,2,3,4,5,6,8,10,12,15,20,24,30,40,60,120}; differences = {1,1,1,1,1,2,2,2,3,5,4,6,10,20,60}; intersection = {1,2,3,4,5,6,10,20,60}, so a(5) = 9.

Links

Crossrefs

Cf. A000142, A027423, A060737, A060738, A060741.

Programs

  • Maple
    f:= proc(n) local D,L;
  D:= numtheory:-divisors(n!);
  L:= sort(convert(D,list));
  nops(convert(L[2..-1]-L[1..-2],set) intersect D);
end proc:
map(f, [$0..34]); # Robert Israel, Jul 03 2017
  • Mathematica
    a[n_ ] := Length[Intersection[Drop[d=Divisors[n! ], 1]-Drop[d, -1], d]]
  • PARI
    a(n) = {my(v = List(), f = n!, d1 = 1, del); fordiv(f, d, if(d > 1, del = d - d1; if(!(f % del), listput(v, del)); d1 = d)); #Set(v);} \\ Amiram Eldar, Jun 15 2024

Formula

a(n) = A060741(n!/2) for n >= 2. - Amiram Eldar, Jun 15 2024

Extensions

Edited by Dean Hickerson, Jan 22 2002
One more term from Robert G. Wilson v, Jan 29 2002
a(33)-a(35) from Robert Israel, Jul 03 2017
a(36)-a(40) from Amiram Eldar, Jun 15 2024

A079178 Number of isomorphism classes of non-anti-associative closed binary operations on a set of order n, listed by class size.

Original entry on oeis.org

0, 2, 6, 3, 10, 78, 3229, 2, 1, 12, 30, 246, 495, 48427, 178914959
Offset: 1

Views

Author

Christian van den Bosch (cjb(AT)cjb.ie), Jan 03 2003

Keywords

Comments

A079178(n)+A079181(n)=A079171(n).
Elements per row: 1,2,4,8,16,30,... (given by A027423, number of positive divisors of n!)
First four rows: 0; 2,6; 3,10,78,3229; 2,1,12,30,246,495,48427,178914959
A079176(x) is equal to the sum of the products of each element in row x of this sequence and the corresponding element of A079210.
The sum of each row x of this sequence is given by A079177(x).

Links

Crossrefs

Cf. A079176, A079177, A079179.

A079200 Number of isomorphism classes of associative non-commutative closed binary operations on a set of order n, listed by class size.

Original entry on oeis.org

0, 0, 2, 0, 2, 0, 4, 6, 2, 0, 0, 4, 5, 0, 46, 73, 2, 0, 0, 0, 4, 0, 0, 8, 0, 2, 36, 0, 43, 2, 473, 1020, 2, 0, 0, 0, 0, 4, 0, 0, 0, 0, 8, 0, 0, 4, 0, 36, 0, 0, 0, 0, 86, 0, 0, 38, 415, 0, 758, 32, 6682, 18426, 2, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8
Offset: 0

Views

Author

Christian van den Bosch (cjb(AT)cjb.ie), Jan 03 2003

Keywords

Comments

Number of elements per row: 1,1,2,4,8,16,30,... (given by A027423, number of positive divisors of n!).

Examples

			Triangle T(n,k) begins:
  0;
  0;
  2, 0;
  2, 0, 4, 6;
  2, 0, 0, 4, 5, 0, 46, 73;
  2, 0, 0, 0, 4, 0, 0, 8, 0, 2, 36, 0, 43, 2, 473, 1020;
  ...

Links

Crossrefs

Row sums give A079199.
Cf. A027423 (row lengths), A079171, A079194, A079175, A079197, A079198, A079199, A079201, A079210.

Formula

A079194(n,k) + A079197(n,k) + T(n,k) + A079201(n,k) = A079171(n,k).
A079198(n) = Sum_{k>=1} T(n,k)*A079210(n,k).
T(n,k) = A079175(n,k) - A079201(n,k). - Andrew Howroyd, Jan 26 2022

Extensions

a(0)=0 prepended and terms a(16) and beyond from Andrew Howroyd, Jan 26 2022
Previous Showing 31-40 of 121 results. Next

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