A022559 -id:A022559 - cp's OEIS frontend

A023986 Sum of exponents of primes in C(4n,2n) - sum of exponents of primes in C(2n,n).

0, 1, 1, 2, 3, 1, 2, 5, 3, 5, 6, 4, 6, 6, 3, 4, 6, 5, 4, 6, 5, 8, 10, 7, 9, 9, 7, 9, 8, 7, 10, 12, 10, 9, 13, 11, 13, 16, 12, 13, 14, 9, 11, 12, 12, 13, 12, 12, 13, 15, 11, 13, 16, 13, 15, 17, 16, 19, 19, 16, 15, 17, 18, 15, 18, 17, 19, 19, 13, 17, 19, 17, 18, 18, 15, 19, 21, 18, 17, 20, 19, 19, 22, 19, 22

Offset: 0

Clark Kimberling

nonn

Charles R Greathouse IV, Table of n, a(n) for n = 0..10000

Cf. A001222, A022559, A023816, A023834.

Cf. A023978, A023979, A023980, A023981, A023982, A023983, A023984, A023985, A023987, A023990, A023991, A023992, A023993.

a[n_] := PrimeOmega[Binomial[4*n, 2*n]] - PrimeOmega[Binomial[2*n, n]]; Array[a, 100, 0] (* Amiram Eldar, Jun 11 2025 *)

a(n) = my(v = binomial(4*n, 2*n)/binomial(2*n, n)); bigomega(numerator(v)) - bigomega(denominator(v)); \\ Michel Marcus, Sep 30 2013

vp(n,p)=my(s);while(n\=p,s+=n);s
a(n)=my(s);forprime(p=2,4*n,s+=vp(4*n,p)-3*vp(2*n,p)+2*vp(n,p)); s \\ Charles R Greathouse IV, Sep 30 2013

From Amiram Eldar, Jun 11 2025: (Start)
a(n) = A023834(n) - A023816(n).
a(n) = A022559(4*n) - 3*A022559(2*n) + 2*A022559(n). (End)

Name clarified, offset changed to 0 and a(0) prepended by Amiram Eldar, Jun 11 2025

A023987 Sum of exponents of primes in C(5n,3n) - sum of exponents of primes in C(3n,2n).

Original entry on oeis.org

0, 1, 2, 0, 2, 4, 2, 3, 5, 4, 5, 2, 2, 7, 5, 5, 7, 7, 7, 3, 9, 11, 3, 5, 6, 8, 11, 7, 9, 12, 10, 10, 10, 12, 11, 10, 13, 14, 13, 13, 14, 15, 14, 5, 8, 11, 8, 9, 9, 12, 14, 13, 16, 20, 12, 14, 17, 14, 19, 12, 17, 19, 17, 15, 15, 21, 20, 16, 19, 23, 16, 21, 21, 23, 22, 17, 22, 26, 24, 24, 26, 26, 24, 22, 23

Offset: 0

Clark Kimberling

nonn

Amiram Eldar, Table of n, a(n) for n = 0..10000

Cf. A001222, A022559, A023819, A023847.

Cf. A023978, A023979, A023980, A023981, A023982, A023983, A023984, A023985, A023986, A023990, A023991, A023992, A023993.

a[n_] := PrimeOmega[Binomial[5*n, 3*n]] - PrimeOmega[Binomial[3*n, 2*n]]; Array[a, 100, 0] (* Amiram Eldar, Jun 11 2025 *)

a(n) = bigomega(binomial(5*n, 3*n)) - bigomega(binomial(3*n, 2*n)); \\ Amiram Eldar, Jun 11 2025

From Amiram Eldar, Jun 11 2025: (Start)
a(n) = A023847(n) - A023819(n).
a(n) = A022559(5*n) - 2*A022559(3*n) + A022559(n). (End)

Name clarified, offset changed to 0 and a(0) prepended by Amiram Eldar, Jun 11 2025

A023990 Sum of exponents of primes in multinomial coefficient M(4n; 2n,n,n) - sum of exponents of primes in M(3n; n,n,n).

Original entry on oeis.org

0, 1, 1, 1, 2, 2, 1, 3, 3, 4, 4, 1, 2, 4, 3, 3, 5, 4, 4, 3, 4, 7, 4, 4, 5, 7, 5, 4, 5, 5, 6, 6, 7, 7, 9, 7, 9, 11, 8, 9, 9, 9, 8, 6, 7, 7, 6, 6, 8, 9, 8, 8, 9, 10, 7, 8, 9, 10, 11, 7, 8, 11, 11, 10, 12, 13, 13, 12, 11, 14, 12, 11, 13, 14, 12, 12, 14, 14, 13, 14, 14, 15, 15, 13, 14, 14, 11, 10, 13, 12, 13, 13, 12, 15

Offset: 0

Clark Kimberling

nonn

Amiram Eldar, Table of n, a(n) for n = 0..10000

Cf. A001222, A000897, A006480, A022559.

Cf. A023978, A023979, A023980, A023981, A023982, A023983, A023984, A023985, A023986, A023987, A023991, A023992, A023993.

a[n_] := PrimeOmega[Multinomial[2*n, n, n]] - PrimeOmega[Multinomial[n, n, n]]; Array[a, 100, 0] (* Amiram Eldar, Jun 11 2025 *)

a(n) = bigomega((4*n)!/((2*n)!*n!^2)) - bigomega((3*n)!/(n!^3)); \\ Amiram Eldar, Jun 11 2025

From Amiram Eldar, Jun 11 2025: (Start)
a(n) = A023980(n) - A023978(n) = A001222(A000897(n)) - A001222(A006480(n)).
a(n) = A022559(4*n) + 2*A022559(n) - A022559(2*n) - A022559(3*n). (End)

Name clarified, offset changed to 0 and a(0) prepended by Amiram Eldar, Jun 11 2025

A023991 Sum of exponents of primes in multinomial coefficient M(3n; n+1,n,n-1).

Original entry on oeis.org

1, 4, 6, 8, 8, 12, 12, 13, 13, 15, 17, 20, 19, 22, 21, 22, 21, 24, 25, 26, 25, 31, 30, 32, 30, 31, 33, 33, 32, 36, 34, 36, 34, 36, 36, 37, 36, 40, 40, 42, 40, 45, 48, 49, 49, 51, 50, 52, 49, 50, 50, 53, 50, 56, 53, 53, 53, 55, 58, 60, 59, 62, 60, 60, 55, 58, 59, 61, 60, 65, 62, 65, 63, 66, 69, 68

Offset: 1

Clark Kimberling

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A001222, A022559, A023978, A076191, A248707.

Cf. A023978, A023979, A023980, A023981, A023982, A023983, A023984, A023985, A023986, A023987, A023990, A023992, A023993.

a[n_] := PrimeOmega[Multinomial[n+1, n, n-1]]; Array[a, 100] (* Amiram Eldar, Jun 11 2025 *)

a(n) = bigomega((3*n)! / ((n-1)!*n!*(n+1)!)); \\ Amiram Eldar, Jun 11 2025

From Amiram Eldar, Jun 11 2025: (Start)
a(n) = A001222(A248707(n)).
a(n) = A022559(3*n) - A022559(n-1) - A022559(n) - A022559(n+1) = A022559(3*n) - 3*A022559(n) - A001222(n+1) + A001222(n) = A023978(n) - A076191(n). (End)

A023992 Sum of exponents of primes in multinomial coefficient M(4n; n+2,2n,n-2).

Original entry on oeis.org

3, 7, 9, 11, 11, 15, 15, 19, 18, 19, 21, 24, 22, 25, 26, 28, 26, 29, 29, 34, 33, 34, 36, 38, 35, 38, 38, 39, 38, 41, 42, 44, 43, 44, 46, 49, 45, 50, 50, 52, 51, 54, 55, 58, 55, 56, 58, 61, 57, 60, 60, 61, 60, 63, 62, 66, 64, 66, 67, 71, 68, 70, 72, 73, 70, 72, 70, 76, 74, 75, 77, 79, 76

Offset: 2

Clark Kimberling

nonn

Amiram Eldar, Table of n, a(n) for n = 2..10000

Cf. A001222, A022559.

Cf. A023978, A023979, A023980, A023981, A023982, A023983, A023984, A023985, A023986, A023987, A023990, A023991, A023993.

a[n_] := PrimeOmega[Multinomial[2*n, n+2, n-2]]; Array[a, 100] (* Amiram Eldar, Jun 11 2025 *)

a(n) = bigomega((4*n)! / ((2*n)!*(n-2)!*(n+2)!)); \\ Amiram Eldar, Jun 11 2025

a(n) = A022559(4*n) - A022559(2*n) - A022559(n+2) - A022559(n-2). - Amiram Eldar, Jun 11 2025

Missing a(1)=1 inserted by Sean A. Irvine, Jun 17 2019

Offset changed to 2 and a(1) removed by Amiram Eldar, Jun 11 2025

A023993 Sum of exponents of primes in multinomial coefficient M(3n; n+2,n-1,n-1).

Original entry on oeis.org

0, 3, 6, 8, 8, 11, 11, 14, 14, 14, 17, 21, 18, 20, 22, 23, 21, 24, 24, 27, 26, 29, 29, 34, 29, 30, 35, 33, 32, 34, 33, 39, 34, 34, 37, 39, 35, 38, 41, 43, 40, 45, 46, 50, 51, 48, 49, 54, 49, 50, 51, 52, 49, 56, 53, 55, 54, 53, 58, 62, 57, 58, 61, 63, 56, 58, 58, 61, 61, 63, 62, 68, 61, 65, 70

Offset: 1

Clark Kimberling

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A001222, A022559.

Cf. A023978, A023979, A023980, A023981, A023982, A023983, A023984, A023985, A023986, A023987, A023990, A023991, A023992.

a[n_] := PrimeOmega[Multinomial[n + 2, n - 1, n - 1]]; Array[a, 100] (* Amiram Eldar, Jun 11 2025 *)

a(n) = bigomega((3*n)! / ((n+2)!*(n-1)!^2)); \\ Amiram Eldar, Jun 11 2025

a(n) = A022559(3*n) - A022559(n+2) - 2*A022559(n-1). - Amiram Eldar, Jun 11 2025

A335407 Number of anti-run permutations of the prime indices of n!.

Original entry on oeis.org

1, 1, 1, 2, 0, 2, 3, 54, 0, 30, 105, 6090, 1512, 133056, 816480, 127209600, 0, 10090080, 562161600, 69864795000, 49989139200, 29593652088000, 382147120555200, 41810689605484800, 4359985823793600, 3025062801079038720, 49052072750637116160, 25835971971637227375360

Offset: 0

Gus Wiseman, Jul 01 2020

nonn

An anti-run is a sequence with no adjacent equal parts.
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
Conjecture: Only vanishes at n = 4 and n = 8.
a(16) = 0. Proof: 16! = 2^15 * m where bigomega(m) = A001222(m) = 13. We can't separate 15 1's with 13 other numbers. - David A. Corneth, Jul 04 2020

			The a(0) = 1 through a(6) = 3 anti-run permutations:
  ()  ()  (1)  (1,2)  .  (1,2,1,3,1)  (1,2,1,2,1,3,1)
               (2,1)     (1,3,1,2,1)  (1,2,1,3,1,2,1)
                                      (1,3,1,2,1,2,1)

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

The version for Mersenne numbers is A335432.
Anti-run compositions are A003242.
Anti-run patterns are counted by A005649.
Permutations of prime indices are A008480.
Anti-runs are ranked by A333489.
Separable partitions are ranked by A335433.
Inseparable partitions are ranked by A335448.
Anti-run permutations of prime indices are A335452.
Strict permutations of prime indices are A335489.
Cf. A056239, A106351, A112798, A114938, A278990, A292884, A325535, A335125.
Factorial numbers: A000142, A001222, A002982, A007489, A022559, A027423, A054991, A108731, A181819, A181821, A325272, A325273, A325617.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[Select[Permutations[primeMS[n!]],!MatchQ[#,{_,x_,x_,_}]&]],{n,0,10}]

\\ See A335452 for count.
a(n)={count(factor(n!)[,2])} \\ Andrew Howroyd, Feb 03 2021

a(n) = A335452(A000142(n)). - Andrew Howroyd, Feb 03 2021

Terms a(14) and beyond from Andrew Howroyd, Feb 03 2021

A337071 Number of strict chains of divisors starting with n!.

Original entry on oeis.org

1, 1, 2, 6, 40, 264, 3776, 40256, 1168000, 34204032, 1107791872, 23233380352, 1486675898368, 38934372315136, 1999103691427840, 132874800979423232, 20506322412604129280, 776179999255323115520, 107455579038104865996800, 4651534843901106606571520, 731092060557632280262082560

Offset: 0

Gus Wiseman, Aug 16 2020

nonn

			The a(1) = 1 through a(3) = 6 chains:
  1  2    6
     2/1  6/1
          6/2
          6/3
          6/2/1
          6/3/1
The a(4) = 40 chains:
  24  24/1   24/2/1   24/4/2/1   24/8/4/2/1
      24/2   24/3/1   24/6/2/1   24/12/4/2/1
      24/3   24/4/1   24/6/3/1   24/12/6/2/1
      24/4   24/4/2   24/8/2/1   24/12/6/3/1
      24/6   24/6/1   24/8/4/1
      24/8   24/6/2   24/8/4/2
      24/12  24/6/3   24/12/2/1
             24/8/1   24/12/3/1
             24/8/2   24/12/4/1
             24/8/4   24/12/4/2
             24/12/1  24/12/6/1
             24/12/2  24/12/6/2
             24/12/3  24/12/6/3
             24/12/4
             24/12/6

A325617 is the maximal case.
A337070 is the version for superprimorials.
A337074 counts the case with distinct prime multiplicities.
A337105 is the case ending with one.
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.
A253249 counts chains of divisors.
Cf. A001013, A001055, A002033, A022559, A124010, A167865, A336941, A336942.

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

a(n) = 2*A337105(n) for n > 1.

a(n) = A067824(n!).

a(19)-a(20) from Alois P. Heinz, Aug 23 2020

A325618 Numbers m such that there exists an integer partition of m whose reciprocal factorial sum is 1.

Original entry on oeis.org

1, 4, 11, 18, 24, 31, 37, 44, 45, 50, 52, 57, 58, 65, 66, 70, 71, 73, 76, 78, 79, 83, 86, 87, 89, 91, 92, 94, 96, 97, 99, 100, 102, 104, 107, 108, 109, 110, 112, 113, 114, 115, 117, 118, 119, 120, 121, 122, 123, 125, 126, 127, 128, 130, 131

Offset: 1

Gus Wiseman, May 13 2019

nonn

The reciprocal factorial sum of an integer partition (y_1,...,y_k) is 1/y_1! + ... + 1/y_k!.

Conjecture: 137 is the greatest integer not in this sequence. - Charlie Neder, May 14 2019

			The sequence of terms together with an integer partition of each whose reciprocal factorial sum is 1 begins:
   1: (1)
   4: (2,2)
  11: (3,3,3,2)
  18: (3,3,3,3,3,3)
  24: (4,4,4,4,3,3,2)
  31: (4,4,4,4,3,3,3,3,3)
  37: (4,4,4,4,4,4,4,4,3,2)
  44: (4,4,4,4,4,4,4,4,3,3,3,3)
  45: (5,5,5,5,5,4,4,4,3,3,2)
  50: (4,4,4,4,4,4,4,4,4,4,4,4,2)

Charlie Neder, Table of n, a(n) for n = 1..1000

Factorial numbers: A000142, A002982, A007489, A011371, A022559, A064986, A115627, A284605, A325616.

Reciprocal factorial sum: A002966, A051908, A058360, A316854, A316855, A325619, A325620, A325621, A325622, A325623, A325624.

a(11)-a(55) from Charlie Neder, May 14 2019

A023826 Sum of exponents in prime-power factorization of C(4n,n).

Original entry on oeis.org

0, 2, 3, 4, 5, 7, 6, 9, 9, 10, 10, 9, 9, 13, 14, 14, 14, 15, 15, 15, 16, 19, 16, 19, 18, 21, 20, 18, 19, 21, 21, 22, 22, 23, 25, 24, 24, 27, 26, 28, 26, 30, 28, 28, 29, 28, 28, 30, 30, 31, 31, 31, 31, 35, 30, 31, 31, 32, 34, 33, 33, 38, 39, 39, 37, 39, 38, 39, 40, 43, 40, 41, 41, 44, 44, 43, 44, 46, 45

Offset: 0

Clark Kimberling

nonn

Also sum of exponents of primes in multinomial coefficient M(4n; n,n,n,n)/M(3n; n,n,n).

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

Cf. A001222, A005810, A022559.

with(numtheory):a:=proc(n) if n=0 then 0 else bigomega(binomial(4*n,n)) fi end: seq(a(n), n=0..78); # Zerinvary Lajos, Apr 11 2008

Join[{0}, Table[Total[FactorInteger[Binomial[4 n, n]][[All, 2]]], {n, 78}]] (* Ivan Neretin, Nov 02 2017 *)

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

From Amiram Eldar, Jun 11 2025: (Start)
a(n) = A001222(A005810(n)).
a(n) = A022559(4*n) - A022559(3*n) - A022559(n). (End)

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

cp's OEIS Frontend

A023986 Sum of exponents of primes in C(4n,2n) - sum of exponents of primes in C(2n,n).

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A023987 Sum of exponents of primes in C(5n,3n) - sum of exponents of primes in C(3n,2n).

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A023990 Sum of exponents of primes in multinomial coefficient M(4n; 2n,n,n) - sum of exponents of primes in M(3n; n,n,n).

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A023991 Sum of exponents of primes in multinomial coefficient M(3n; n+1,n,n-1).

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A023992 Sum of exponents of primes in multinomial coefficient M(4n; n+2,2n,n-2).

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A023993 Sum of exponents of primes in multinomial coefficient M(3n; n+2,n-1,n-1).

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A335407 Number of anti-run permutations of the prime indices of n!.

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A337071 Number of strict chains of divisors starting with n!.