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.

A336420 Irregular triangle read by rows where T(n,k) is the number of divisors of the n-th superprimorial A006939(n) with distinct prime multiplicities and k prime factors counted with multiplicity.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 1, 3, 2, 5, 2, 1, 1, 1, 4, 3, 11, 7, 7, 10, 5, 2, 1, 1, 1, 5, 4, 19, 14, 18, 37, 25, 23, 15, 23, 10, 5, 2, 1, 1, 1, 6, 5, 29, 23, 33, 87, 70, 78, 74, 129, 84, 81, 49, 39, 47, 23, 10, 5, 2, 1, 1, 1, 7, 6, 41, 34, 52, 165, 144, 183, 196, 424, 317, 376, 325, 299, 431, 304, 261, 172, 129, 81, 103, 47, 23, 10, 5, 2, 1, 1
Offset: 0

Views

Author

Gus Wiseman, Jul 25 2020

Keywords

Comments

A number's prime signature (row n of A124010) is the sequence of positive exponents in its prime factorization, so a number has distinct prime multiplicities iff all the exponents in its prime signature are distinct.
The n-th superprimorial or Chernoff number is A006939(n) = Product_{i = 1..n} prime(i)^(n - i + 1).
T(n,k) is also the number of length-n vectors 0 <= v_i <= i summing to k whose nonzero values are all distinct.

Examples

			Triangle begins:
  1
  1  1
  1  2  1  1
  1  3  2  5  2  1  1
  1  4  3 11  7  7 10  5  2  1  1
  1  5  4 19 14 18 37 25 23 15 23 10  5  2  1  1
The divisors counted in row n = 4 are:
  1  2  4     8   16   48   144   432  2160  10800  75600
     3  9    12   24   72   360   720  3024
     5  25   18   40   80   400  1008
     7       20   54  108   504  1200
             27   56  112   540  2800
             28  135  200   600
             45  189  675   756
             50            1350
             63            1400
             75            4725
            175

Crossrefs

A000110 gives row sums.
A000124 gives row lengths.
A000142 counts divisors of superprimorials.
A006939 lists superprimorials or Chernoff numbers.
A008278 is the version counting only distinct prime factors.
A008302 counts divisors of superprimorials by bigomega.
A022915 counts permutations of prime indices of superprimorials.
A076954 can be used instead of A006939.
A130091 lists numbers with distinct prime multiplicities.
A146291 counts divisors by bigomega.
A181796 counts divisors with distinct prime multiplicities.
A181818 gives products of superprimorials.
A317829 counts factorizations of superprimorials.
A336417 counts perfect-power divisors of superprimorials.
A336498 counts divisors of factorials by bigomega.
A336499 uses factorials instead superprimorials.
Cf. A000005, A001222, A008278, A027423, A071625, A124010, A327498, A336419, A336421, A336426, A336500, A336568.

Programs

  • Mathematica
    chern[n_]:=Product[Prime[i]^(n-i+1),{i,n}];
Table[Length[Select[Divisors[chern[n]],PrimeOmega[#]==k&&UnsameQ@@Last/@FactorInteger[#]&]],{n,0,5},{k,0,n*(n+1)/2}]

A336499 Irregular triangle read by rows where T(n,k) is the number of divisors of n! with distinct prime multiplicities and a total of k prime factors, counted with multiplicity.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 0, 1, 2, 1, 2, 1, 1, 3, 1, 3, 2, 0, 1, 3, 2, 5, 3, 3, 2, 1, 1, 4, 2, 7, 4, 4, 3, 2, 0, 1, 4, 2, 7, 4, 5, 7, 7, 6, 3, 2, 0, 1, 4, 2, 8, 8, 9, 10, 11, 11, 7, 8, 5, 2, 0, 1, 4, 3, 11, 8, 11, 16, 16, 15, 15, 15, 13, 9, 6, 3, 1, 1, 5, 3, 14, 10, 13, 21, 21, 20, 19, 21, 18, 13, 9, 5, 2, 0
Offset: 0

Views

Author

Gus Wiseman, Aug 03 2020

Keywords

Comments

Row lengths are A022559(n) + 1.

Examples

			Triangle begins:
  1
  1
  1  1
  1  2  0
  1  2  1  2  1
  1  3  1  3  2  0
  1  3  2  5  3  3  2  1
  1  4  2  7  4  4  3  2  0
  1  4  2  7  4  5  7  7  6  3  2  0
  1  4  2  8  8  9 10 11 11  7  8  5  2  0
  1  4  3 11  8 11 16 16 15 15 15 13  9  6  3  1
  1  5  3 14 10 13 21 21 20 19 21 18 13  9  5  2  0
  1  5  3 14 10 14 25 23 27 24 30 28 28 25 20 16 11  5  2  0
Row n = 7 counts the following divisors:
  1  2  4  8   16  48   144  720   {}
     3  9  12  24  72   360  1008
     5     18  40  80   504
     7     20  56  112
           28
           45
           63

Crossrefs

A000720 is column k = 1.
A022559 gives row lengths minus one.
A056172 appears to be column k = 2.
A336414 gives row sums.
A336420 is the version for superprimorials.
A336498 is the version counting all divisors.
A336865 is the generalization to non-factorials.
A336866 lists indices of rows with a final 1.
A336867 lists indices of rows with a final 0.
A336868 gives the final terms in each row.
A000110 counts divisors of superprimorials with distinct prime exponents.
A008302 counts divisors of superprimorials by number of prime factors.
A130091 lists numbers with distinct prime exponents.
A181796 counts divisors with distinct prime exponents.
A327498 gives the maximum divisor of n with distinct prime exponents.
Cf. A000005, A001222, A008278, A098859, A118914, A124010, A146291, A336422, A336500.
Factorial numbers: A000142, A002982, A027423, A048656, A048742, A054991, A071626, A336425, A336617.

Programs

  • Mathematica
    Table[Length[Select[Divisors[n!],PrimeOmega[#]==k&&UnsameQ@@Last/@FactorInteger[#]&]],{n,0,6},{k,0,PrimeOmega[n!]}]

A103774 Number of ways to write n! as product of squarefree numbers.

Original entry on oeis.org

1, 1, 2, 2, 6, 10, 42, 42, 82, 204, 1196, 1556, 10324, 34668, 104948, 104964, 873540, 1309396, 11855027, 25238220, 91193575, 453628255, 5002616219, 5902762219, 21142729523, 122981607092, 189706055368, 547296181656, 7291700021313, 14330422534833, 202498591157970
Offset: 1

Views

Author

Reinhard Zumkeller, Feb 15 2005

Keywords

Comments

a(n) = A050320(A000142(n)).
From Gus Wiseman, Aug 20 2020: (Start)
Also the number of set multipartitions (multisets of sets) of the multiset of prime factors of n!. For example, The a(2) = 1 through a(6) = 10 set multipartitions are:
{1} {12} {1}{1}{12} {1}{1}{123} {1}{1}{12}{123}
{1}{2} {1}{1}{1}{2} {1}{12}{13} {1}{12}{12}{13}
{1}{1}{1}{23} {1}{1}{1}{12}{23}
{1}{1}{2}{13} {1}{1}{1}{2}{123}
{1}{1}{3}{12} {1}{1}{2}{12}{13}
{1}{1}{1}{2}{3} {1}{1}{3}{12}{12}
{1}{1}{1}{1}{2}{23}
{1}{1}{1}{2}{2}{13}
{1}{1}{1}{2}{3}{12}
{1}{1}{1}{1}{2}{2}{3}
(End)

Examples

			n=5, 5! = 1*2*3*4*5 = 120 = 2 * 2 * 2 * 3 * 5: a(5)=#{2*2*2*3*5,2*2*2*15,2*2*6*5,2*2*30,2*2*3*10,2*6*10}=6.

Links

Crossrefs

A103775 is the strict case.
A157612 is the case of superprimorials.
A001055 counts factorizations.
A045778 counts strict factorizations.
A048656 counts squarefree divisors of factorials.
A050320 counts factorizations into squarefree numbers.
A050326 counts strict factorizations into squarefree numbers.
A076716 counts factorizations of factorials.
A089259 counts set multipartitions of integer partitions.
A116540 counts normal set multipartitions.
A157612 counts strict factorizations of factorials.
Cf. A000110, A005117, A008480, A124010, A318360.
Factorial numbers: A000142, A007489, A022559, A027423, A071626, A325272, A325617, A336498.

Programs

  • Mathematica
    sub[w_, e_] := Block[{v=w}, v[[e]]--; v]; ric[w_, k_] := ric[w, k] = If[Max[w] == 0, 1, Block[{e, s, p = Flatten@ Position[Sign@w, 1]}, s = Select[ Prepend[#, First@p] & /@ Subsets[Rest@p], Total[1/2^#] <= k &]; Sum[ric[sub[w, e], Total[1/2^e]], {e, s}]]]; a[n_] := ric[ Sort[ Last /@ FactorInteger[n!]], 1]; Array[a, 22] (* Giovanni Resta, Sep 30 2019 *)

Extensions

a(17)-a(18) from Amiram Eldar, Sep 30 2019
a(19)-a(31) from Giovanni Resta, Sep 30 2019

A103775 Number of ways to write n! as product of distinct squarefree numbers.

Original entry on oeis.org

1, 1, 2, 0, 1, 0, 3, 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, 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: 1

Views

Author

Reinhard Zumkeller, Feb 15 2005

Keywords

Comments

From Gus Wiseman, Aug 23 2020: (Start)
Also the number of set-systems (sets of sets) whose multiset union is the multiset of prime factors of n!. For example, the a(1) = 1 through a(7) = 3 set-systems (empty columns indicated by dots) are:
0 {1} {1,2} . {1},{1,2},{1,3} . {1},{1,2},{1,3},{1,2,4}
{1},{2} {1},{1,2},{1,4},{1,2,3}
{1},{2},{1,2},{1,3},{1,4}
(End)

Examples

			n=7, 7! = 1*2*3*4*5*6*7 = 5040 = 2*2*2*2*3*3*5*7: a(7) = #{2*3*6*10*14, 2*6*10*42, 2*6*14*30} = 3.

Links

Crossrefs

A103774 is the non-strict version.
A337073 is the version for superprimorials, with non-strict version A337072.
A001055 counts factorizations.
A045778 counts strict factorizations.
A048656 counts squarefree divisors of factorials.
A050320 counts factorizations into squarefree numbers.
A050326 counts strict factorizations into squarefree numbers.
A050342 counts set-systems by total sum.
A076716 counts factorizations of factorials.
A116539 counts set-systems covering an initial interval.
A157612 counts strict factorizations of factorials.
Cf. A000110, A005117, A008480, A089259, A116540, A124010, A318360.
Factorial numbers: A000142, A007489, A022559, A027423, A071626, A325272, A325617, A336498.

Programs

  • Mathematica
    yst[n_]:=yst[n]=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[yst[n/d],Min@@#>d&]],{d,Select[Rest[Divisors[n]],SquareFreeQ]}]];
Table[Length[yst[n!]],{n,15}] (* Gus Wiseman, Aug 21 2020 *)

Formula

a(n) = 0 for n > 7;
a(n) = A050326(A000142(n)).

A336940 Number of odd divisors of n!.

Original entry on oeis.org

1, 1, 1, 2, 2, 4, 6, 12, 12, 20, 30, 60, 72, 144, 216, 336, 336, 672, 864, 1728, 2160, 3200, 4800, 9600, 10560, 14784, 22176, 28224, 35280, 70560, 86400, 172800, 172800, 245760, 368640, 497664, 559872, 1119744, 1679616, 2363904, 2626560, 5253120, 6451200, 12902400, 16128000
Offset: 0

Views

Author

Gus Wiseman, Aug 23 2020

Keywords

Examples

			The a(1) = 1 through a(8) = 12 divisors:
  1  1  1  1  1   1   1    1
        3  3  3   3   3    3
              5   5   5    5
              15  9   7    7
                  15  9    9
                  45  15   15
                      21   21
                      35   35
                      45   45
                      63   63
                      105  105
                      315  315

Links

Crossrefs

A049606 gives the maximum among these divisors, with quotient A060818.
A337257 is the even version.
A000265 gives the maximum odd divisor of n.
A001227 counts odd divisors.
A183063 counts even divisors.
Cf. A000005, A001013, A001055, A006939, A113474, A124010, A253249.
Factorial numbers: A000142, A022559, A027423 (divisors), A048656, A071626, A076716 (factorizations), A325272, A325273, A325617, A336414, A336498.

Programs

  • Mathematica
    Table[Length[Select[Divisors[n!],OddQ]],{n,0,15}]
  • PARI
    a(n) = sumdiv(n!, d, d%2); \\ Michel Marcus, Aug 24 2020
  • PARI
    a(n) = numdiv(prod(k=1, n, k >> valuation(k, 2))); \\ Michel Marcus, Aug 27 2020

Formula

a(n) = A001227(n!).
a(n) = A000005(A049606(n)).
a(n) + A337257(n) = A027423(n) = A000005(n!).
From Seiichi Manyama, Aug 27 2020: (Start)
If p is odd prime, a(p) = 2 * a(p-1).
a(n) = A027423(n) / A113474(n) for n > 0. (End)

Extensions

a(36)-a(44) from Seiichi Manyama, Aug 26 2020

A337257 Number of even divisors of n!.

Original entry on oeis.org

0, 0, 1, 2, 6, 12, 24, 48, 84, 140, 240, 480, 720, 1440, 2376, 3696, 5040, 10080, 13824, 27648, 38880, 57600, 91200, 182400, 232320, 325248, 510048, 649152, 882000, 1764000, 2246400, 4492800, 5356800, 7618560, 11796480, 15925248
Offset: 0

Views

Author

Gus Wiseman, Aug 23 2020

Keywords

Examples

			The a(2) = 1 through a(5) = 12 divisors:
  2  2  2   2
     6  4   4
        6   6
        8   8
        12  10
        24  12
            20
            24
            30
            40
            60
            120

Links

Crossrefs

A336940 is the odd version.
A000265 gives the maximum odd divisor of n.
A001227 counts odd divisors.
A183063 counts even divisors.
Cf. A000005, A001013, A001055, A006939, A049606, A124010, A253249.
Factorial numbers: A000142, A022559, A027423 (divisors), A048656, A071626, A076716 (factorizations), A325272, A325273, A325617, A336414, A336498.

Programs

  • Mathematica
    Table[Length[Select[Divisors[n!],EvenQ]],{n,0,15}]
  • PARI
    a(n) = sumdiv(n!, d, !(d%2)); \\ Michel Marcus, Aug 24 2020

Formula

a(n) = A183063(n!).
A336940(n) + a(n) = A027423(n) = A000005(n!).
Showing 1-6 of 6 results.

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

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