A064350 -id:A064350 - cp's OEIS frontend

A025487 Least integer of each prime signature A124832; also products of primorial numbers A002110.

1, 2, 4, 6, 8, 12, 16, 24, 30, 32, 36, 48, 60, 64, 72, 96, 120, 128, 144, 180, 192, 210, 216, 240, 256, 288, 360, 384, 420, 432, 480, 512, 576, 720, 768, 840, 864, 900, 960, 1024, 1080, 1152, 1260, 1296, 1440, 1536, 1680, 1728, 1800, 1920, 2048, 2160, 2304, 2310

Offset: 1

David W. Wilson

All numbers of the form 2^k1*3^k2*...*p_n^k_n, where k1 >= k2 >= ... >= k_n, sorted.
A111059 is a subsequence. - Reinhard Zumkeller, Jul 05 2010
Choie et al. (2007) call these "Hardy-Ramanujan integers". - Jean-François Alcover, Aug 14 2014
The exponents k1, k2, ... can be read off Abramowitz & Stegun p. 831, column labeled "pi".
For all such sequences b for which it holds that b(n) = b(A046523(n)), the sequence which gives the indices of records in b is a subsequence of this sequence. For example, A002182 which gives the indices of records for A000005, A002110 which gives them for A001221 and A000079 which gives them for A001222. - Antti Karttunen, Jan 18 2019
The prime signature corresponding to a(n) is given in row n of A124832. - M. F. Hasler, Jul 17 2019

			The first few terms are 1, 2, 2^2, 2*3, 2^3, 2^2*3, 2^4, 2^3*3, 2*3*5, ...

Franklin T. Adams-Watters, Table of n, a(n) for n = 1..10001 (first 291 terms from Will Nicholes)
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972.
Kevin Broughan, Equivalents of the Riemann Hypothesis, Vol. 1: Arithmetic Equivalents, Cambridge University Press, 2017. See section 8.2, "Hardy-Ramanujan Numbers".
YoungJu Choie, Nicolas Lichiardopol, Pieter Moree and Patrick Solé, On Robin's criterion for the Riemann hypothesis, Journal de théorie des nombres de Bordeaux, Vol. 19, No. 2 (2007), pp. 357-372. See section 5, p. 367.
Asaf Cohen Antonir and Asaf Shapira, An Elementary Proof of a Theorem of Hardy and Ramanujan (2022). arXiv:2207.09410 [math.NT]
Michael De Vlieger, Relations of A025487 to A002110, A002182, and A002201.
Steven R. Finch, Errata and Addenda to Mathematical Constants, arXiv:2001.00578 [math.HO], 2020, pp. 9-10.
G. H. Hardy and S. Ramanujan, Asymptotic formulae for the distribution of integers of various types, Proc. London Math. Soc, Ser. 2, Vol. 16 (1917), pp. 112-132. Also published in the book Collected Papers of Srinivasa Ramanujan, Chelsea, 1962, pages 245-261.
Jeffery Kline, On the eigenstructure of sparse matrices related to the prime number theorem, Linear Algebra and its Applications (2020) Vol. 584, 409-430.
L. B. Richmond, Asymptotic results for partitions (I) and the distribution of certain integers, Journal of Number Theory, Vol. 8, No. 4 (1976), pp. 372-389. See page 388.

Subsequence of A055932, A191743, and A324583.
Cf. A025488, A051282, A036041, A051466, A061394, A124832, A161360, A166469, A181815, A181817, A283980, A306802, A322584, A322585 (characteristic function), A329897, A329898, A329899, A329900, A329904, A330683.
Cf. A085089, A101296 (left inverses).
Equals range of values taken by A046523.
Cf. A178799 (first differences), A247451 (squarefree kernel), A146288 (number of divisors).
Subsequences of this sequence include: A000079, A000142, A000400, A001013, A001813, A002110, A002182, A005179, A006939, A025527, A056836, A061742, A064350, A066120, A087980, A097212, A097213, A111059, A119840, A119845, A126098, A129912, A140999, A166338, A166470, A166472, A166473, A166475, A167448, A168262, A168263, A168264, A179215, A181555, A181804, A181806, A181809, A181818, A181822, A181823, A181824, A181825, A181826, A181827, A182763, A182862, A182863, A212170, A220264, A220423, A250269, A250270, A260633, A266047, A284456, A300357, A304938, A329894, A330687; also A037019 and A330681 (when sorted), possibly also A289132.
Rearrangements of this sequence include A036035, A059901, A063008, A077569, A085988, A086141, A087443, A108951, A181821, A181822, A322827, A329886, A329887.
Cf. also array A124832 (row n = prime signature of a(n)) and A304886, A307056.

import Data.Set (singleton, fromList, deleteFindMin, union)
a025487 n = a025487_list !! (n-1)
a025487_list = 1 : h [b] (singleton b) bs where
   (_ : b : bs) = a002110_list
   h cs s xs'@(x:xs)
     | m <= x    = m : h (m:cs) (s' `union` fromList (map (* m) cs)) xs'
     | otherwise = x : h (x:cs) (s  `union` fromList (map (* x) (x:cs))) xs
     where (m, s') = deleteFindMin s
-- Reinhard Zumkeller, Apr 06 2013

isA025487 := proc(n)
    local pset,omega ;
    pset := sort(convert(numtheory[factorset](n),list)) ;
    omega := nops(pset) ;
    if op(-1,pset) <> ithprime(omega) then
        return false;
    end if;
    for i from 1 to omega-1 do
        if padic[ordp](n,ithprime(i)) < padic[ordp](n,ithprime(i+1)) then
            return false;
        end if;
    end do:
    true ;
end proc:
A025487 := proc(n)
    option remember ;
    local a;
    if n = 1 then
        1 ;
    else
        for a from procname(n-1)+1 do
            if isA025487(a) then
                return a;
            end if;
        end do:
    end if;
end proc:
seq(A025487(n),n=1..100) ; # R. J. Mathar, May 25 2017

PrimeExponents[n_] := Last /@ FactorInteger[n]; lpe = {}; ln = {1}; Do[pe = Sort@PrimeExponents@n; If[ FreeQ[lpe, pe], AppendTo[lpe, pe]; AppendTo[ln, n]], {n, 2, 2350}]; ln (* Robert G. Wilson v, Aug 14 2004 *)
(* Second program: generate all terms m <= A002110(n): *)
f[n_] := {{1}}~Join~
  Block[{lim = Product[Prime@ i, {i, n}],
   ww = NestList[Append[#, 1] &, {1}, n - 1], dec},
   dec[x_] := Apply[Times, MapIndexed[Prime[First@ #2]^#1 &, x]];
   Map[Block[{w = #, k = 1},
      Sort@ Prepend[If[Length@ # == 0, #, #[[1]]],
        Product[Prime@ i, {i, Length@ w}] ] &@ Reap[
         Do[
          If[# < lim,
             Sow[#]; k = 1,
             If[k >= Length@ w, Break[], k++]] &@ dec@ Set[w,
             If[k == 1,
               MapAt[# + 1 &, w, k],
               PadLeft[#, Length@ w, First@ #] &@
                 Drop[MapAt[# + Boole[i > 1] &, w, k], k - 1] ]],
           {i, Infinity}] ][[-1]]
] &, ww]]; Sort[Join @@ f@ 13] (* Michael De Vlieger, May 19 2018 *)

isA025487(n)=my(k=valuation(n,2),t);n>>=k;forprime(p=3,default(primelimit),t=valuation(n,p);if(t>k,return(0),k=t);if(k,n/=p^k,return(n==1))) \\ Charles R Greathouse IV, Jun 10 2011

factfollow(n)={local(fm, np, n2);
  fm=factor(n); np=matsize(fm)[1];
  if(np==0,return([2]));
  n2=n*nextprime(fm[np,1]+1);
  if(np==1||fm[np,2]Franklin T. Adams-Watters, Dec 01 2011 */

is(n) = {if(n==1, return(1)); my(f = factor(n));  f[#f~, 1] == prime(#f~) && vecsort(f[, 2],,4) == f[, 2]} \\ David A. Corneth, Feb 14 2019

upto(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]))))) \\ M. F. Hasler, Jul 17 2019

\\ For fast generation of large number of terms, use this program:
A283980(n) = {my(f=factor(n)); prod(i=1, #f~, my(p=f[i, 1], e=f[i, 2]); if(p==2, 6, nextprime(p+1))^e)}; \\ From A283980
A025487list(e) = { my(lista = List([1, 2]), i=2, u = 2^e, t); while(lista[i] != u, if(2*lista[i] <= u, listput(lista,2*lista[i]); t = A283980(lista[i]); if(t <= u, listput(lista,t))); i++); vecsort(Vec(lista)); }; \\ Returns a list of terms up to the term 2^e.
v025487 = A025487list(101);
A025487(n) = v025487[n];
for(n=1,#v025487,print1(A025487(n), ", ")); \\ Antti Karttunen, Dec 24 2019

def sharp_primorial(n): return sloane.A002110(prime_pi(n))
N = 2310
nmax = 2^floor(log(N,2))
sorted([j for j in (prod(sharp_primorial(t[0])^t[1] for k, t in enumerate(factor(n))) for n in (1..nmax)) if j <= N])
# Giuseppe Coppoletta, Jan 26 2015

What can be said about the asymptotic behavior of this sequence? - Franklin T. Adams-Watters, Jan 06 2010
Hardy & Ramanujan prove that there are exp((2 Pi + o(1))/sqrt(3) * sqrt(log x/log log x)) members of this sequence up to x. - Charles R Greathouse IV, Dec 05 2012
From Antti Karttunen, Jan 18 & Dec 24 2019: (Start)
A085089(a(n)) = n.
A101296(a(n)) = n [which is the first occurrence of n in A101296, and thus also a record.]
A001221(a(n)) = A061395(a(n)) = A061394(n).
A007814(a(n)) = A051903(a(n)) = A051282(n).
a(A101296(n)) = A046523(n).
a(A306802(n)) = A002182(n).
a(n) = A108951(A181815(n)) = A329900(A181817(n)).
If A181815(n) is odd, a(n) = A283980(a(A329904(n))), otherwise a(n) = 2*a(A329904(n)).
(End)
Sum_{n>=1} 1/a(n) = Product_{n>=1} 1/(1 - 1/A002110(n)) = A161360. - Amiram Eldar, Oct 20 2020

Offset corrected by Matthew Vandermast, Oct 19 2008

Minor correction by Charles R Greathouse IV, Sep 03 2010

A157704 G.f.s of the z^p coefficients of the polynomials in the GF3 denominators of A156927.

Original entry on oeis.org

1, 1, 5, 32, 186, 132, 10, 56, 2814, 17834, 27324, 11364, 1078, 10, 48, 17988, 494720, 3324209, 7526484, 6382271, 2004296, 203799, 4580, 5, 16, 72210, 7108338, 146595355, 1025458635, 2957655028, 3828236468

Offset: 0

Johannes W. Meijer, Mar 07 2009

The formula for the PDGF3(z;n) polynomials in the GF3 denominators of A156927 can be found below.
The general structure of the GFKT3(z;p) that generate the z^p coefficients of the PDGF3(z; n) polynomials can also be found below. The KT3(z;p) polynomials in the numerators of the GFKT3(z; p) have a nice symmetrical structure.
The sequence of the number of terms of the first few KT3(z;p) polynomials is 1, 2, 4, 7, 10, 13, 14, 17, 20, 23, 26, 29, 32, 34, 36, 39, 42. The differences of this sequence and that of the number of terms of the KT4(z;p), see A157705, follow a simple pattern.
A Maple algorithm that generates relevant GFKT3(z;p) information can be found below.

			Some PDGF3 (z;n) are:
  PDGF3(z;n=3) = (1-z)*(1-2*z)^4*(1-3*z)^7*(1-4*z)^10
  PDGF3(z;n=4) = (1-z)*(1-2*z)^4*(1-3*z)^7*(1-4*z)^10*(1-5*z)^13
The first few GFKT3's are:
  GFKT3(z;p=0) = 1/(1-z)
  GFKT3(z;p=1) = -(5*z+1)/(1-z)^4
  GFKT3(z;p=2) = z*(32+186*z+132*z^2+10*z^3)/(1-z)^7
Some KT3(z,p) polynomials are:
  KT3(z;p=2) = 32+186*z+132*z^2+10*z^3
  KT3(z;p=3) = 56+2814*z+17834*z^2+27324*z^3+11364*z^4+1078*z^5+10*z^6

Originator sequence A156927.
See A002414 for the z^1 coefficients and A157707 for the z^2 coefficients divided by 2.
Row sums equal A064350 and those of A157705.
Cf. A157702, A157703, A157705.

p:=2; fn:=sum((-1)^(n1+1)*binomial(3*p+1,n1) *a(n-n1),n1=1..3*p+1): fk:=rsolve(a(n) = fn,a(k)): for n2 from 0 to 3*p+1 do fz(n2):=product((1-(k+1)*z)^(1+3*k), k=0..n2): a(n2):= coeff(fz(n2),z,p); end do: b:=n-> a(n): seq(b(n), n=0..3*p+1); a(n)=fn; a(k)=sort(simplify(fk)); GFKT3(p):=sum((fk)*z^k, k=0..infinity); q3:=ldegree((numer(GFKT3(p)))): KT3(p):=sort((-1)^(p)*simplify((GFKT3(p)*(1-z)^(3*p+1))/z^q3),z, ascending);

PDGF3(z;n) = Product_{k=0..n} (1-(k+1)*z)^(1+3*k) with n = 1, 2, 3, ...
GFKT3(z;p) = (-1)^(p)*(z^q3)*KT3(z, p)/(1-z)^(3*p+1) with p = 0, 1, 2, ...
The recurrence relation for the z^p coefficients a(n) is a(n) = Sum_{k=1..3*p+1} (-1)^(k+1)*binomial(3*p + 1, k)*a(n-k) with p = 0, 1, 2, ... .

A157705 G.f.s of the z^p coefficients of the polynomials in the GF4 denominators of A156933.

Original entry on oeis.org

1, 1, 3, 2, 18, 128, 171, 42, 1, 22, 1348, 11738, 26293, 17693, 3271, 115, 13, 6122, 228986, 2070813, 6324083, 7397855, 3361536, 544247, 24590, 155, 3, 17248, 2413434, 67035224, 612026240, 2274148882

Offset: 0

Johannes W. Meijer, Mar 07 2009

The formula for the PDGF4(z;n) polynomials in the GF4 denominators of A156933 can be found below.
The general structure of the GFKT4(z;p) that generate the z^p coefficients of the PDGF4(z;n) polynomials can also be found below. The KT4(z;p) polynomials in the numerators of the GFKT4(z;p) have a nice symmetrical structure.
The sequence of the number of terms of the first few KT4(z;p) polynomials is 1, 3, 5, 7, 10, 13, 15, 18, 20, 23, 26, 29, 32, 34, 37, 40, 42. The differences of this sequence and that of the number of terms of the KT3(z;p), see A157704, follow a simple pattern.
A Maple algorithm that generates relevant GFKT4(z;p) information can be found below.

			Some PDGF4 (z;n) are:
  PDGF4(z; n=3) = (1-7*z)*(1-5*z)^4*(1-3*z)^7*(1-z)^10
  PDGF4(z; n=4) = (1-9*z)*(1-7*z)^4*(1-5*z)^7*(1-3*z)^10*(1-z)^13
The first few GFKT4's are:
  GFKT4(z;p=0) = 1/(1-z)
  GFKT4(z;p=1) = -(1+3*z+2*z^2)/(1-z)^4
  GFKT4(z;p=2) = z*(18+128*z+171*z^2+42*z^3+z^4)/(1-z)^7
Some KT4(z,p) polynomials are:
  KT4(z;p=2) = 18+128*z+171*z^2+42*z^3+z^4
  KT4(z;p=3) = 22+1348*z+11738*z^2+26293*z^3+17693*z^4+3271*z^5+115*z^6

Originator sequence A156933.
See A081436 for the z^1 coefficients and A157708 for the z^2 coefficients.
Row sums equal A064350 and those of A157704.
Cf. A157702, A157703, A157704.

p:=2; fn:=sum((-1)^(n1+1)*binomial(3*p+1,n1) *a(n-n1),n1=1..3*p+1): fk:=rsolve(a(n) = fn,a(k)): for n2 from 0 to 3*p+1 do fz(n2):=product((1-(2*n2+1-(2*k))*z)^(3*k+1), k=0..n2): a(n2):= coeff(fz(n2),z,p): end do: b:=n-> a(n): seq(b(n), n=0..3*p+1); a(n)=fn; a(k)= sort (simplify(fk)); GFKT4(p):=sum((fk)*z^k,k=0..infinity); q4:=ldegree((numer (GFKT4(p)))): KT4(p):=sort((-1)^(p)*simplify((GFKT4(p)*(1-z)^(3*p+1))/z^q4),z, ascending);

PDGF4(z;n) = Product_{k=0..n} (1-(2*n+1-2*k)*z)^(3*k+1) with n = 1, 2, 3, ...
GFKT4(z;p) = (-1)^(p)*(z^q4)*KT4(z, p)/(1-z)^(3*p+1) with p = 0, 1, 2, ...
The recurrence relation for the z^p coefficients a(n) is a(n) = Sum_{k=1..3*p+1} (-1)^(k+1)*binomial(3*p + 1, k)*a(n-k) with p = 0, 1, 2, ... .

A250270 Products of terms of A003418.

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 16, 24, 32, 36, 48, 60, 64, 72, 96, 120, 128, 144, 192, 216, 240, 256, 288, 360, 384, 420, 432, 480, 512, 576, 720, 768, 840, 864, 960, 1024, 1152, 1296, 1440, 1536, 1680, 1728, 1920, 2048, 2160, 2304, 2520, 2592, 2880, 3072, 3360, 3456

Offset: 1

Matthew Vandermast, Dec 16 2014

nonn

Includes all factorials and Jordan-Polya numbers, since n! = Product_{i = 1..n} A003418(floor(n/i)) for positive n.

			720 = 2*6*60 = 12*60. Since 2, 6, 12 and 60 are all terms of A003418, 720 is a term of this sequence.

Michel Marcus, Table of n, a(n) for n = 1..5000

Range of values of A253139. Subsequences include A000142, A001013, A001813, A025527, A064350, A166338, A250569.

Subsequence of A025487.

f(n) = lcm(vector(n, i, i)); \\ A003418
mul(x,y) = x*y;
lista(nn) = {my(v = vector(nn, k, f(k)), lim = f(nn+1), ok = 0, nv); while (!ok,  nv = select(x->(xMichel Marcus, May 09 2021

A202946 a(n+1) = 6A060544(n)a(n).

Original entry on oeis.org

3, 18, 1080, 181440, 59875200, 32691859200, 26676557107200, 30411275102208000, 46164315605151744000, 90020415430045900800000, 219289731987591814348800000, 652606242395073239502028800000

Offset: 1

John M. Campbell, Dec 26 2011

Sums of coefficients from (3n+1)th moments of binomial(m,k)*binomial(2m,k): see Maple code below.

			The evaluation of sum(k=0..n, k^7*binomial(n,k)*binomial(2*n,k)) involves the polynomial 32*n^7+96*n^6-336*n^5-360*n^4+1020*n^3-42*n^2-455*n+63, the sum of the coefficients of which is 18 = a(2).

Eric W. Weisstein, MathWorld: Binomial Sums
Index to divisibility sequences

Cf. A060544, A064350.

with(PolynomialTools); polyn := proc (q) options operator, arrow; 3^q*Pi*GAMMA(2*n)*(sum(k^q*binomial(n, k)*binomial(2*n, k), k = 0 .. n))/(27^n*sqrt(3)*GAMMA(n-floor((1/3)*q+1/3)+2/3)*GAMMA(n-floor((1/3)*q)+1/3)) end proc; coefl := proc (q) options operator, arrow; CoefficientList(expand(polyn(q)), n) end proc; coe := proc (j, h) options operator, arrow; coefl(j)[h] end proc; seq(sum(coe(3*r+1, k), k = 1 .. 5*r+1), r = 1 .. 8) ;

print1(a=3);for(n=2,10,print1(", ",a*=27*n*(n-3)+60)) \\ Charles R Greathouse IV, Dec 26 2011

a(n) = (1/18)*27^n*Gamma(n-1/3)*Gamma(n-2/3)*sqrt(3)/Pi.

A202948 a(n+1) = 3A136016a(n).

Original entry on oeis.org

-3, -72, -7560, -1814400, -778377600, -523069747200, -506854585036800, -669048052248576000, -1154107890128793600000, -2520571632041285222400000, -6797981691615346244812800000

Offset: 1

John M. Campbell, Dec 26 2011

Sums of coefficients from (3n+2)th moments of binomial(m,k)*binomial(2m,k): see Maple code below.

			The evaluation of sum(k^8 binomial(n,k) binomial(2n,k), k=0..n) involves the polynomial 64n^10 + 192n^9 - 1344n^8 - 1056n^7 + 8256n^6 - 3696n^5 - 9940n^4 + 7551n^3 + 348n^2 - 507n + 60, the sum of the coefficients of which is -72=a(2).

Eric W. Weisstein, MathWorld: Binomial Sums

Cf. A136016, A064350.

with(PolynomialTools); polyn := proc (q) options operator, arrow; 3^q*Pi*GAMMA(2*n)*(sum(k^q*binomial(n, k)*binomial(2*n, k), k = 0 .. n))/(27^n*sqrt(3)*GAMMA(n-floor((1/3)*q+1/3)+2/3)*GAMMA(n-floor((1/3)*q)+1/3)) end proc; coefl := proc (q) options operator, arrow; CoefficientList(expand(polyn(q)), n) end proc; coe := proc (j, h) options operator, arrow; coefl(j)[h] end proc; seq(sum(coe(3*r+2, k), k = 1 .. 5*r+3), r = 1 .. 8) ;

print1(a=-3);for(n=2,20,print1(", ",a*=27*n*(n-2)+24)) \\ Charles R Greathouse IV, Dec 27 2011

a(n)=-(1/6)*27^n*GAMMA(n-1/3)*GAMMA(n+1/3)*sqrt(3)/Pi.

A001525 a(n) = (3n)!/(3!n!).

Original entry on oeis.org

1, 60, 10080, 3326400, 1816214400, 1482030950400, 1689515283456000, 2564684200286208000, 5001134190558105600000, 12182762888199545241600000, 36255902355281846639001600000, 129433571408356192501235712000000, 545950804200446419970212233216000000

Offset: 1

N. J. A. Sloane

nonn

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..100

Equals A064350/3!, row sums of A157704/3! and row sums of A157705/3!.

Table[(3 n)!/(3! n!), {n, 20}] (* Harvey P. Dale, May 25 2011 *)

More terms from Harvey P. Dale, May 25 2011

A066387 Triangle T(n,m) (1<=m<=n) giving number of maps f:N -> N such that f^m(X)=X+n for all natural numbers X.

Original entry on oeis.org

1, 1, 2, 1, 0, 6, 1, 12, 0, 24, 1, 0, 0, 0, 120, 1, 120, 360, 0, 0, 720, 1, 0, 0, 0, 0, 0, 5040, 1, 1680, 0, 20160, 0, 0, 0, 40320, 1, 0, 60480, 0, 0, 0, 0, 0, 362880, 1, 30240, 0, 0, 1814400, 0, 0, 0, 0, 3628800, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 39916800

Offset: 1

Floor van Lamoen, Dec 23 2001

			Triangle T(n,m) begins:
  1;
  1,    2;
  1,    0,   6;
  1,   12,   0,    24;
  1,    0,   0,     0, 120;
  1,  120, 360,     0,   0, 720;
  1,    0,   0,     0,   0,   0, 5040;
  1, 1680,   0, 20160,   0,   0,    0, 40320;
  ...

Vincenzo Librandi, Rows n = 1..100, flattened
A. Heinis, R. Jeurissen and L. Kamstra, Problem 18 and solution, Nieuw Arch. Wisk. 5/2 (2001) 380.

Row sums give A057625.
Main diagonal gives A000142.
m-section of column m=2-4 (for n>0) gives: A001813, A064350, A166338.

t[n_, m_] /; Divisible[n, m] := n!/(n/m)!; t[, ] = 0; Flatten[Table[t[n, m], {n, 1, 11}, {m, 1, n}]] (* Jean-François Alcover, Nov 29 2011 *)

T(n,m) = n!/(n/m)! if m|n, T(n,m) = 0 otherwise.

A066991 Square array read by descending antidiagonals of number of ways of dividing n*k labeled items into k unlabeled orders with n items in each order.

Original entry on oeis.org

1, 1, 2, 1, 12, 6, 1, 120, 360, 24, 1, 1680, 60480, 20160, 120, 1, 30240, 19958400, 79833600, 1814400, 720, 1, 665280, 10897286400, 871782912000, 217945728000, 239500800, 5040, 1, 17297280, 8892185702400, 20274183401472000

Offset: 1

Henry Bottomley, Feb 01 2002

T(p,k) = (pk)!/k! is divisible by p^k but not p^(k+1) for p prime; e.g., T(3,4) = 3^4*11*10*8*7*5*4*2*1 = 19958400.

			The array begins:
  n\k|   1        2             3                   4  ...
  --------------------------------------------------------
   1 |   1,       1,            1,                  1, ...
   2 |   2,      12,          120,               1680, ...
   3 |   6,     360,        60480,           19958400, ...
   4 |  24,   20160,     79833600,       871782912000, ...
   5 | 120, 1814400, 217945728000, 101370917007360000, ...
  ...

Paolo Xausa, Table of n, a(n) for n = 1..820 (antidiagonals 1..40 of the array, flattened).

Rows include A000012, A001813, A064350.
Columns include A000142, A002674, A065961.
Cf. A060538, A060539, A060540.

Table[((n-k+1)*k)!/k!, {n, 10}, {k, n, 1, -1}] (* Paolo Xausa, Feb 19 2024 *)

T(n,k) = (n*k)!/k!.

Edited by Paolo Xausa, Feb 19 2024

cp's OEIS Frontend

A025487 Least integer of each prime signature A124832; also products of primorial numbers A002110.

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A157704 G.f.s of the z^p coefficients of the polynomials in the GF3 denominators of A156927.

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A157705 G.f.s of the z^p coefficients of the polynomials in the GF4 denominators of A156933.

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A250270 Products of terms of A003418.

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A202946 a(n+1) = 6*A060544(n)*a(n).

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A202948 a(n+1) = 3*A136016*a(n).

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A001525 a(n) = (3n)!/(3!n!).

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A202946 a(n+1) = 6A060544(n)a(n).

A202948 a(n+1) = 3A136016a(n).