A027423 -id:A027423 - cp's OEIS frontend

A079171 Number of isomorphism classes of closed binary operations (groupoids) on a set of order n, listed by class size.

1, 4, 6, 3, 12, 78, 3237, 2, 1, 14, 30, 275, 495, 48810, 178932325

Offset: 1

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

Elements per row: 1,2,4,8,16,30,... (given by A027423, number of positive divisors of n!)
A002489(n) is equal to the sum of the products of each element in row n of this sequence and the corresponding element of A079210.
The sum of each row n is given by A001329(n).

			First four rows:
  1;
  4, 6;
  3, 12, 78, 3237;
  2, 1, 14, 30, 275, 495, 48810, 178932325.

C. van den Bosch, Closed binary operations on small sets
Index entries for sequences related to groupoids

Cf. A002489, A001329. a(n, A027423(n)) = A030245(n).

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

Gus Wiseman, Jul 25 2020

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.

			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

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.

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}]

A337105 Number of strict chains of divisors from n! to 1.

Original entry on oeis.org

1, 1, 1, 3, 20, 132, 1888, 20128, 584000, 17102016, 553895936, 11616690176, 743337949184, 19467186157568, 999551845713920, 66437400489711616, 10253161206302064640, 388089999627661557760, 53727789519052432998400, 2325767421950553303285760, 365546030278816140131041280

Offset: 0

Gus Wiseman, Aug 17 2020

nonn

			The a(4) = 20 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

Andrew Howroyd, Table of n, a(n) for n = 0..200

A325617 is the maximal case.
A336941 is the version for superprimorials.
A337104 counts the case with distinct prime multiplicities.
A337071 is the case not necessarily ending with 1.
A000005 counts divisors.
A000142 lists factorial numbers.
A001055 counts factorizations.
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.
A253249 counts chains of divisors.
A336423 counts chains using A130091, with maximal case A336569.
A336942 counts chains using A130091 from A006939(n) to 1.
Cf. A001013, A001055, A022559, A124010, A167865, A337070, A337074.

b:= proc(n) option remember; 1 +
      add(b(d), d=numtheory[divisors](n) minus {n})
    end:
a:= n-> ceil(b(n!)/2):
seq(a(n), n=0..14);  # Alois P. Heinz, Aug 23 2020

chnsc[n_]:=Prepend[Join@@Table[Prepend[#,n]&/@chnsc[d],{d,DeleteCases[Divisors[n],1|n]}],{n}];
Table[Length[chnsc[n!]],{n,0,5}]

a(n) = A337071(n)/2 for n > 1.

a(n) = A074206(n!).

a(19)-a(20) from Alois P. Heinz, Aug 22 2020

A060776 Smaller central (median) divisor of n!.

Original entry on oeis.org

1, 1, 2, 4, 10, 24, 70, 192, 576, 1890, 6300, 21600, 78848, 294840, 1143072, 4572288, 18849600, 79968000, 348566400, 1559376000, 7147140000, 33522128640, 160758097500, 787652812800, 3938264064000, 20080974513600, 104348440350000, 552160113120000

Offset: 1

Labos Elemer, Apr 26 2001

nonn

Factorial splitting: write n! = x*y with x <= y and x maximal; sequence gives value of x. Inequality "x < y" gives the same sequence, except that a(1) is not defined.

Between this d and its complementary divisor, the integer part of square root of n! is situated; for n=6: {24,26,30}. - Nathaniel Johnston, Jun 25 2011

			Divisors of 6!=720 are {1,2,3,4,5,6,...,24,30,...,360,720}. a(6)=24, the 15th divisor from 30 divisors of 720.

Max Alekseyev, Table of n, a(n) for n = 1..140 (first 59 terms from Oleg Terentyev)
Jean-Marie De Koninck and William Verreault, Arithmetic functions at factorial arguments, Publications de l'Institut Mathematique, Vol. 115, No. 129 (2024), pp. 45-76.

Cf. A027423, A055226, A000196, A000142, A000005, A060777, A033676.

Table[ Part[ Divisors[ w! ], Floor[ DivisorSigma[ 0, n! ]/2 ] ], {w, a, b} ]

a(n) = if (n==1, 1, my(d=divisors(n!)); d[#d\2]); \\ Michel Marcus, Sep 16 2018

a(n) = n!/A060777(n). - David Wasserman, Jun 15 2002
a(n) = A033676(A000142(n)). - Pontus von Brömssen, Jul 15 2023
Sum_{k=1..n} a(k) = sqrt(n!) * (1 + O(1/n^c)), where c < 1 is a positive constant (De Koninck and Verreault, 2024, p. 48, Theorem 2.1). - Amiram Eldar, Dec 10 2024

More terms from David Wasserman, Jun 15 2002

a(27)-a(32) from M. F. Hasler, Sep 20 2011

A060777 Larger central (or median) divisor of n!.

Original entry on oeis.org

1, 2, 3, 6, 12, 30, 72, 210, 630, 1920, 6336, 22176, 78975, 295680, 1144000, 4576000, 18869760, 80061696, 348986880, 1560176640, 7148445696, 33530112000, 160813154304, 787718131200, 3938590656000, 20083261440000, 104351051284480, 552173794099200, 2973519499493376, 16286922357866496, 90680032493568000, 512971179263262720

Offset: 1

Labos Elemer, Apr 26 2001

nonn

Factorial splitting: write n! = x*y with x <= y and x maximal; sequence gives value of y. Inequality "x < y" gives the same sequence, except that a(1) is not defined.

The integer part of square root of n! (A055226(n)) is situated between x and y.

			Divisors of 6!=720 are {1, 2, 3, 4, 5, 6, ..., 24, 30, ..., 360, 720}. a(6)=30, the 16th one from the 30 divisors of 720.

Max Alekseyev, Table of n, a(n) for n = 1..140
Jean-Marie De Koninck and William Verreault, Arithmetic functions at factorial arguments, Publications de l'Institut Mathematique, Vol. 115, No. 129 (2024), pp. 45-76.

Cf. A027423, A055226, A000196, A000142, A000005, A060776, A061057, A033677.

Table[ Part[ Divisors[ w! ], 1+Floor[ DivisorSigma[ 0, n! ]/2 ] ], {w, a, b} ]

a(n) = A033677(A000142(n)). - Pontus von Brömssen, Jul 15 2023

Sum_{k=1..n} a(k) = sqrt(n!) * (1 + O(1/n^c)), where c < 1 is a positive constant (De Koninck and Verreault, 2024, p. 48, Theorem 2.1). - Amiram Eldar, Dec 10 2024

More terms from Don Reble, Dec 13 2001

A131688 Decimal expansion of the constant Sum_{k>=1} log(k + 1) / (k * (k + 1)).

Original entry on oeis.org

1, 2, 5, 7, 7, 4, 6, 8, 8, 6, 9, 4, 4, 3, 6, 9, 6, 3, 0, 0, 0, 9, 8, 9, 9, 8, 3, 0, 4, 9, 5, 8, 8, 1, 5, 2, 8, 5, 1, 1, 5, 4, 0, 8, 9, 0, 5, 0, 8, 8, 8, 4, 8, 6, 8, 9, 7, 7, 5, 4, 0, 8, 3, 3, 5, 2, 2, 5, 4, 9, 9, 9, 4, 8, 9, 3, 7, 4, 4, 9, 3, 4, 9, 7, 0, 7, 9, 0, 4, 7, 3, 1, 5, 0, 1, 9, 0, 9, 7, 8, 2, 4, 5, 4, 8

Offset: 1

R. J. Mathar, Sep 14 2007

Given A131385(n) = Product_{k=1..n} floor((n+k)/k), then limit A131385(n+1)/A131385(n) = exp(c), where c = this constant. - Paul D. Hanna, Nov 26 2012

Closely related to A085361 (the exponent in the definition of A085291). - Yuriy Sibirmovsky, Sep 04 2016

			1.257746886944369630009899830495881528511540890508884868977540833522...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, page 62. [Jean-François Alcover, Mar 21 2013]

G. C. Greubel, Table of n, a(n) for n = 1..1000
Khristo N. Boyadzhiev, A special constant and series with zeta values and harmonic numbers, arXiv:1903.11141 [math.NT], 2019.
Mark W. Coffey, Series of zeta values, the Stieltjes constants and a sum S_gamma(n), arXiv:math-ph/0706.0345, 2007-2009, eq (38a).
Paul Erdős, S. W. Graham, Aleksandar Ivic and Carl Pomerance, On the number of divisors of n!, Analytic Number Theory, Volume 138, Progress in Mathematics pp 337-355.
Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 538.
Sofia Kalpazidou, Khintchine's constant for Lüroth representation, Journal of Number Theory, Volume 29, Issue 2, June 1988, Pages 196-205.

Cf. A000005, A002210, A027423, A075887, A131385, A244109, A001620, A085361, A345682.

SetDefaultRealField(RealField(100)); L:=RiemannZeta(); (&+[(-1)^(n+1)*Evaluate(L,n+1)/n: n in [1..10^3]]); // G. C. Greubel, Nov 15 2018

evalf(sum((-1)^(n+1)*Zeta(n+1)/n, n=1..infinity), 120); # Vaclav Kotesovec, Dec 11 2015
evalf(Sum(-Zeta(1, k), k = 2..infinity), 120); # Vaclav Kotesovec, Jun 18 2021

Sum[ -Zeta'[1 + k], {k, 1, Infinity}] (* Vladimir Reshetnikov, Dec 28 2008 *)
Integrate[EulerGamma/x + PolyGamma[0, 1+x]/x, {x, 0, 1}] // N[#, 105]& // RealDigits[#][[1]]& (* or *) Integrate[x*Log[x]/((1-x)*Log[1-x]), {x, 0, 1}] // N[#, 105]& // RealDigits[#][[1]]& (* Jean-François Alcover, Feb 04 2013 *)
$MaxExtraPrecision = 200; NIntegrate[HarmonicNumber[t]/t, {t, 0, 1}, WorkingPrecision -> 105] (* Yuriy Sibirmovsky, Sep 04 2016 *)
digits = 120; RealDigits[NSum[(-1)^(n + 1)*Zeta[n + 1]/n, {n,1,Infinity}, NSumTerms -> 20*digits, WorkingPrecision -> 10*digits, Method -> "AlternatingSigns"], 10, digits][[1]] (* G. C. Greubel, Nov 15 2018 *)

PARI
```
sumalt(s=1, (-1)^(s+1)/s*zeta(s+1) )
```

suminf(k=2, -zeta'(k)) \\ Vaclav Kotesovec, Jun 17 2021

numerical_approx(sum((-1)^(k+1)*zeta(k+1)/k for k in [1..1000]), digits=100) # G. C. Greubel, Nov 15 2018

Equals Sum_{s>=1} (-1)^(s+1)*zeta(s+1)/s.
Equals Sum_{k>=1} -zeta'(1 + k), where Zeta' is the derivative of the Riemann zeta function. - Vladimir Reshetnikov, Dec 28 2008
Equals Sum_{s>=1} log(1+1/s)/s. - Jean-François Alcover, Mar 26 2013
Equals Integral_{t=0..1} H(t)/t dt. Compare to A001620 = Integral_{t=0..1} H(t) dt. Where H(t) are generalized harmonic numbers. - Yuriy Sibirmovsky, Sep 04 2016
Equals lim_{n->oo} log(d(n!))*log(n)/n, where d(n) is the number of divisors of n (A000005) (Erdős et al., 1996). - Amiram Eldar, Nov 07 2020

Extended to 105 digits by Jean-François Alcover, Feb 04 2013

A336415 Number of divisors of n! with equal prime multiplicities.

Original entry on oeis.org

1, 1, 2, 4, 6, 10, 13, 21, 24, 28, 33, 49, 53, 85, 94, 100, 104, 168, 173, 301, 307, 317, 334, 590, 595, 603, 636, 642, 652, 1164, 1171, 2195, 2200, 2218, 2283, 2295, 2301, 4349, 4478, 4512, 4519, 8615, 8626, 16818, 16836, 16844, 17101, 33485, 33491, 33507, 33516, 33582

Offset: 0

Gus Wiseman, Jul 22 2020

nonn

A number k has "equal prime multiplicities" (or is "uniform") iff its prime signature is constant, meaning that k is a power of a squarefree number.

			The a(n) uniform divisors of n for n = 1, 2, 6, 8, 30, 36 are the columns:
  1  2  6  8  30  36
     1  3  6  15  30
        2  4  10  16
        1  3   8  15
           2   6  10
           1   5   9
               4   8
               3   6
               2   5
               1   4
                   3
                   2
                   1
In 20!, the multiplicity of the third prime (5) is 4 but the multiplicity of the fourth prime (7) is 2. Hence there are 2^3 - 1 = 3 divisors with all exponents 3 (we subtract |{1}| = 1 from that count as 1 has no exponent 3). - _David A. Corneth_, Jul 27 2020

David A. Corneth, Table of n, a(n) for n = 0..9999
Jon Maiga, Computer-generated formulas for A336415, Sequence Machine.

The version for distinct prime multiplicities is A336414.
The version for nonprime perfect powers is A336416.
Uniform partitions are counted by A047966.
Uniform numbers are A072774, with nonprime terms A182853.
Numbers with distinct prime multiplicities are A130091.
Divisors with distinct prime multiplicities are counted by A181796.
Maximum divisor with distinct prime multiplicities is A327498.
Uniform divisors are counted by A327527.
Maximum uniform divisor is A336618.
1st differences are given by A048675.
Cf. A000005, A000961, A001222, A001597, A007916, A112798, A124010, A327499.
Factorial numbers: A000142, A007489, A022559, A027423, A048656, A071626, A108731, A325272, A325273, A325617.

Table[Length[Select[Divisors[n!],SameQ@@Last/@FactorInteger[#]&]],{n,0,15}]

a(n) = sumdiv(n!, d, my(ex=factor(d)[,2]); (#ex==0) || (vecmin(ex) == vecmax(ex))); \\ Michel Marcus, Jul 24 2020

a(n) = {if(n<2, return(1)); my(f = primes(primepi(n)), res = 1, t = #f); f = vector(#f, i, val(n, f[i])); for(i = 1, f[1], while(f[t] < i, t--; ); res+=(1<David A. Corneth, Jul 27 2020

a(n) = A327527(n!).

Terms a(31) and onwards from David A. Corneth, Jul 27 2020

A336417 Number of perfect-power divisors of superprimorials A006939.

Original entry on oeis.org

1, 1, 2, 5, 15, 44, 169, 652, 3106, 15286, 89933, 532476, 3698650, 25749335, 204947216, 1636097441, 14693641859, 132055603656, 1319433514898, 13186485900967, 144978145009105, 1594375302986404, 19128405558986057, 229508085926717076, 2983342885319348522

Offset: 0

Gus Wiseman, Jul 24 2020

nonn

A number is a perfect power iff it is 1 or its prime exponents (signature) are not relatively prime.

The n-th superprimorial number is A006939(n) = Product_{i = 1..n} prime(i)^(n - i + 1).

			The a(0) = 1 through a(4) = 15 divisors:
  1  2  12  360  75600
-------------------------
  1  1   1    1      1
         4    4      4
              8      8
              9      9
             36     16
                    25
                    27
                    36
                   100
                   144
                   216
                   225
                   400
                   900
                  3600

Andrew Howroyd, Table of n, a(n) for n = 0..200

A000325 is the uniform version.
A076954 can be used instead of A006939.
A336416 gives the same for factorials instead of superprimorials.
A000217 counts prime power divisors of superprimorials.
A000961 gives prime powers.
A001597 gives perfect powers, with complement A007916.
A006939 gives superprimorials or Chernoff numbers.
A022915 counts permutations of prime indices of superprimorials.
A091050 counts perfect power divisors.
A181818 gives products of superprimorials.
A294068 counts factorizations using perfect powers.
A317829 counts factorizations of superprimorials.
Cf. A000005, A027423, A090630, A118914, A124010, A203025, A251753, A327527, A336419, A336420, A336421, A336426.

chern[n_]:=Product[Prime[i]^(n-i+1),{i,n}];
perpouQ[n_]:=Or[n==1,GCD@@FactorInteger[n][[All,2]]>1];
Table[Length[Select[Divisors[chern[n]],perpouQ]],{n,0,5}]

a(n) = {1 + sum(k=2, n, moebius(k)*(1 - prod(i=1, n, 1 + i\k)))} \\ Andrew Howroyd, Aug 30 2020

a(n) = A091050(A006939(n)).

a(n) = 1 + Sum_{k=2..n} mu(k)*(1 - Product_{i=1..n} 1 + floor(i/k)). - Andrew Howroyd, Aug 30 2020

Terms a(10) and beyond from Andrew Howroyd, Aug 30 2020

A048742 a(n) = n! - (n-th Bell number).

Original entry on oeis.org

0, 0, 0, 1, 9, 68, 517, 4163, 36180, 341733, 3512825, 39238230, 474788003, 6199376363, 86987391878, 1306291409455, 20912309745853, 355604563226196, 6401691628921841, 121639267666626943, 2432850284018404628, 51090467301893283249, 1123996221061869232677

Offset: 0

N. J. A. Sloane

nonn

Number of permutations of [n] which have at least one cycle that has at least one inversion when written with its smallest element in the first position. Example: a(4)=9 because we have (1)(243), (1432), (142)(3), (132)(4), (1342), (1423), (1243), (143)(2) and (1324). - Emeric Deutsch, Apr 29 2008
Number of permutations of [n] having consecutive runs of increasing elements with initial elements in increasing order. a(4) = 9: `124`3, `13`24, `134`2, `14`23, `14`3`2, `2`14`3, `24`3`1, `3`14`2, `4`13`2. - Alois P. Heinz, Apr 27 2016
From Gus Wiseman, Aug 11 2020: (Start)
Also the number of divisors of the superfactorial A006939(n - 1) without distinct prime multiplicities. For example, the a(4) = 9 divisors together with their prime signatures are the following. Note that A076954 can be used here instead of A006939.
    6: (1,1)
   10: (1,1)
   15: (1,1)
   30: (1,1,1)
   36: (2,2)
   60: (2,1,1)
   90: (1,2,1)
  120: (3,1,1)
  180: (2,2,1)
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..450
Index entries for sequences related to factorial numbers

A000110 lists Bell numbers.
A000142 lists factorial numbers.
A006939 lists superprimorials or Chernoff numbers.
A181796 counts divisors with distinct prime multiplicities.
A336414 counts divisors of n! with distinct prime multiplicities.
Cf. A000005, A022915, A027423, A076954, A095149, A130091, A336420, A336421.

with(combinat): seq(factorial(n)-bell(n),n=0..21); # Emeric Deutsch, Apr 29 2008

Table[n! - BellB[n], {n, 0, 40}] (* Vladimir Joseph Stephan Orlovsky, Jul 06 2011 *)

[factorial(m) - bell_number(m) for m in range(23)]  # Zerinvary Lajos, Jul 06 2008

a(n) = A000142(n) - A000110(n).

E.g.f.: 1/(1-x) - exp(exp(x)-1). - Alois P. Heinz, Apr 27 2016

A079197 Number of isomorphism classes of non-associative commutative closed binary operations on a set of order n, listed by class size.

Original entry on oeis.org

0, 0, 1, 1, 4, 5, 107, 0, 0, 0, 5, 0, 28, 488, 43389

Offset: 1

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

A079194(n)+A079197(n)+A079200(n)+A079201(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; 0,1; 1,4,5,107; 0,0,0,5,0,28,488,43389
A079195(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 A079196(x).

C. van den Bosch, Closed binary operations on small sets
Index entries for sequences related to groupoids

Cf. A079194, A079198, A079199, A079200, A079201.

cp's OEIS Frontend

A079171 Number of isomorphism classes of closed binary operations (groupoids) on a set of order n, listed by class size.

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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.

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A337105 Number of strict chains of divisors from n! to 1.

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A060776 Smaller central (median) divisor of n!.

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A060777 Larger central (or median) divisor of n!.

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A131688 Decimal expansion of the constant Sum_{k>=1} log(k + 1) / (k * (k + 1)).

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A336415 Number of divisors of n! with equal prime multiplicities.

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