A022559 -id:A022559 - cp's OEIS frontend

A023834 Sum of exponents in prime-power factorization of C(4n,2n).

0, 2, 3, 5, 6, 6, 7, 11, 9, 11, 12, 12, 13, 15, 14, 15, 15, 16, 15, 18, 17, 20, 22, 22, 22, 23, 22, 23, 22, 23, 25, 28, 25, 25, 29, 28, 28, 32, 30, 32, 31, 30, 31, 34, 34, 34, 34, 36, 35, 37, 34, 36, 38, 38, 38, 40, 38, 41, 42, 42, 40, 44, 46, 44, 43, 43, 44, 46, 42, 46, 47, 47, 46, 48, 47, 50, 51, 50

Offset: 0

Clark Kimberling

nonn

Ivan Neretin, Table of n, a(n) for n = 0..10000

Cf. A001222, A001448, A023816, A022559.

Join[{0}, Table[Total[FactorInteger[Binomial[4 n, 2 n]][[All, 2]]], {n, 77}]] (* Ivan Neretin, Oct 26 2017 *)

a(n) = bigomega(binomial(4*n, 2*n)); \\ Amiram Eldar, Jun 11 2025

From Amiram Eldar, Jun 11 2025: (Start)
a(n) = A001222(A001448(n)).
a(n) = A023816(2*n).
a(n) = A022559(4*n) - 2*A022559(2*n). (End)

Offset changed to 0 and a(0) prepended by Amiram Eldar, Jun 11 2025

A303279 Expansion of (1/(1 - x)^2) * Sum_{p prime, k>=1} x^(p^k)/(1 - x^(p^k)).

Original entry on oeis.org

0, 1, 3, 7, 12, 19, 27, 38, 51, 66, 82, 101, 121, 143, 167, 195, 224, 256, 289, 325, 363, 403, 444, 489, 536, 585, 637, 692, 748, 807, 867, 932, 999, 1068, 1139, 1214, 1290, 1368, 1448, 1532, 1617, 1705, 1794, 1886, 1981, 2078, 2176, 2279, 2384, 2492, 2602, 2715, 2829, 2947, 3067

Offset: 1

Ilya Gutkovskiy, Apr 20 2018

nonn

Sum of exponents in prime-power factorization of product of first n factorials (A000178).

Partial sums of A022559.

			a(5) = 12 because 2!*3!*4!*5! = 2^8*3^3*5 and 8 + 3 + 1 = 12.

Daniel Suteu, Table of n, a(n) for n = 1..10000
Jean-Marie De Koninck and William Verreault, Arithmetic functions at factorial arguments, Publications de l'Institut Mathématique, Vol. 115, No. 129 (2024), pp. 45-76.
Eric Weisstein's World of Mathematics, Barnes G-Function.
Eric Weisstein's World of Mathematics, Superfactorial.
Index entries for sequences computed from exponents in factorization of n.
Index entries for sequences related to factorial numbers.

Cf. A000178, A001222, A022559, A083342, A303281.

Half of row sums of triangle A356718.

b:= proc(n) option remember; `if`(n<1, [0$2],
      (p-> p+[numtheory[bigomega](n), p[1]])(b(n-1)))
    end:
a:= n-> b(n+1)[2]:
seq(a(n), n=1..55);  # Alois P. Heinz, Oct 07 2021

nmax = 55; Rest[CoefficientList[Series[1/(1 - x)^2 Sum[Boole[PrimePowerQ[k]] x^k/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x]]
Table[PrimeOmega[BarnesG[n + 2]], {n, 55}]
Table[ Sum[ PrimeOmega@ j, {k, n}, {j, k}], {n, 55}]

a(n) = my(t=0); sum(k=1, n, t+=bigomega(k)); \\ Daniel Suteu, Jan 17 2019

a(n) = (n^2/2) * (log(log(n)) + c + O(1/log(n))), where c = A083342 (De Koninck and Verreault, 2024, p. 49, eq. (3.1)). - Amiram Eldar, Dec 10 2024

a(n) = (1/2) * Sum_{k=0..n} A356718(n,k). - Dario T. de Castro, Feb 19 2025

A023819 Sum of exponents in prime-power factorization of C(3n,n).

Original entry on oeis.org

0, 1, 2, 4, 4, 4, 6, 8, 6, 7, 8, 11, 11, 11, 11, 12, 10, 12, 11, 15, 13, 13, 18, 18, 17, 16, 17, 19, 17, 18, 19, 22, 18, 18, 20, 21, 19, 21, 22, 23, 22, 21, 23, 28, 27, 27, 28, 30, 27, 28, 26, 28, 29, 28, 31, 32, 29, 31, 31, 35, 32, 33, 35, 34, 31, 30, 31, 34, 31, 32, 35, 36, 33, 34, 35, 38, 37, 36, 36

Offset: 0

Clark Kimberling

nonn

Equivalently, sum of exponents of primes in multinomial coefficient M(3n; n,n,n)/C(2n,n).

Ivan Neretin, Table of n, a(n) for n = 0..10000

Cf. A001222, A005809, A022559.

with(numtheory):(combinat):a:=proc(n) if n=0 then 0 else bigomega(binomial(3*n,n)) fi end: seq(a(n), n=0..78); # Zerinvary Lajos, Apr 11 2008

Join[{0}, Table[Total[FactorInteger[Binomial[3 n, n]][[All, 2]]], {n, 78}]] (* Ivan Neretin, Nov 02 2017 *)

a(n) = bigomega(binomial(3*n, n)); \\ Amiram Eldar, Jun 11 2025

From Amiram Eldar, Jun 11 2025: (Start)
a(n) = A001222(A005809(n)).
a(n) = A022559(3*n) - A022559(2*n) - A022559(n). (End)

Edited by N. J. A. Sloane at the suggestion of R. J. Mathar, May 31 2008

Offset changed to 0 and a(0) prepended by Amiram Eldar, Jun 11 2025

A023827 Sum of exponents in prime-power factorization of C(4n,n-1).

Original entry on oeis.org

0, 3, 3, 6, 4, 7, 8, 10, 9, 11, 8, 11, 10, 15, 14, 16, 13, 16, 14, 18, 15, 17, 17, 21, 20, 21, 19, 20, 18, 22, 21, 26, 21, 26, 24, 27, 23, 26, 28, 28, 28, 30, 26, 30, 27, 29, 29, 33, 30, 33, 30, 33, 30, 33, 31, 33, 31, 33, 32, 36, 35, 39, 39, 42, 37, 40, 38, 41, 40, 42, 40, 44, 41, 45, 44, 46, 44, 46

Offset: 1

Clark Kimberling

nonn

Ivan Neretin, Table of n, a(n) for n = 1..10000

Cf. A001222, A004331, A022559, A023826.

Join[{0},Table[Total[Transpose[FactorInteger[Binomial[4n,n-1]]] [[2]]],{n,2,80}]] (* Harvey P. Dale, Jun 13 2013 *)
a[n_] := PrimeOmega[Binomial[4*n, n-1]]; Array[a, 100] (* Amiram Eldar, Jun 11 2025 *)

a(n) = bigomega(binomial(4*n, n-1)); \\ Amiram Eldar, Jun 11 2025

From Amiram Eldar, Jun 11 2025: (Start)
a(n) = A001222(A004331(n)). - Amiram Eldar, Jun 11 2025
a(n) = A022559(4*n) - A022559(3*n+1) - A022559(n-1).
a(n) = A023826(n) - A001222(3*n+1) + A001222(n). (End)

A325509 Number of factorizations of n! into factorial numbers > 1.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 1, 2, 2, 3, 1, 2, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 0

Gus Wiseman, May 08 2019

nonn

			n = 10:
  (6*120*5040)
  (720*5040)
  (3628800)
n = 16:
  (2*2*2*2*1307674368000)
  (2*120*87178291200)
  (20922789888000)
n = 24:
  (2*2*6*25852016738884976640000)
  (24*25852016738884976640000)
  (620448401733239439360000)

Cf. A000142, A011371, A022559, A034876, A034878, A076934, A109100, A109101, A109102, A115627, A135291, A322583, A325272, A325273, A325276, A325508, A325510, A325511.

facs[n_,u_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d,u],Min@@#>=d&]],{d,Intersection[u,Rest[Divisors[n]]]}]];
Table[Length[facs[n!,Rest[Array[#!&,n]]]],{n,15}]

a(n) = 1 + A034876(n).

More terms from Alois P. Heinz, May 08 2019

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

Gus Wiseman, Aug 13 2020

nonn

			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

Andrew Howroyd, Table of n, a(n) for n = 0..25

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.

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}]

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

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

a(n) = A074206(A006939(n)).

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

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

Gus Wiseman, Aug 03 2020

nonn

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

			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}

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.

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

Gus Wiseman, Aug 16 2020

nonn

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

			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

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.

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}]

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

Gus Wiseman, Aug 23 2020

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

			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

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.

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

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!]}]

A023816 Sum of exponents in prime-power factorization of C(2n,n).

Original entry on oeis.org

0, 1, 2, 3, 3, 5, 5, 6, 6, 6, 6, 8, 7, 9, 11, 11, 9, 11, 11, 12, 12, 12, 12, 15, 13, 14, 15, 14, 14, 16, 15, 16, 15, 16, 16, 17, 15, 16, 18, 19, 17, 21, 20, 22, 22, 21, 22, 24, 22, 22, 23, 23, 22, 25, 23, 23, 22, 22, 23, 26, 25, 27, 28, 29, 25, 26, 25, 27, 29, 29, 28, 30, 28, 30, 32, 31, 30, 32, 32, 33

Offset: 0

Clark Kimberling

nonn

Ivan Neretin, Table of n, a(n) for n = 0..10000

Cf. A000984, A001222, A022559.

with(numtheory):a:=proc(n) if n=0 then 0 else bigomega(binomial(2*n,n)) fi end: seq(a(n), n=0..79); # Zerinvary Lajos, Apr 11 2008

Join[{0},Table[Total[FactorInteger[Binomial[2 n, n]][[All, 2]]], {n, 79}]] (* Ivan Neretin, Oct 26 2017 *)

a(n) = bigomega(binomial(2*n, n)); \\ Amiram Eldar, Jun 11 2025

From Amiram Eldar, Jun 11 2025: (Start)
a(n) = A001222(A000984(n)).
a(n) = A022559(2*n) - 2*A022559(n). (End)

Offset changed to 0 and a(0) prepended by Amiram Eldar, Jun 11 2025

cp's OEIS Frontend

A023834 Sum of exponents in prime-power factorization of C(4n,2n).

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A303279 Expansion of (1/(1 - x)^2) * Sum_{p prime, k>=1} x^(p^k)/(1 - x^(p^k)).

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A023819 Sum of exponents in prime-power factorization of C(3n,n).

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A023827 Sum of exponents in prime-power factorization of C(4n,n-1).

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A325509 Number of factorizations of n! into factorial numbers > 1.

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

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A336496 Products of superfactorials (A000178).

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