A076934 -id:A076934 - cp's OEIS frontend

A071626 Number of distinct exponents in the prime factorization of n!.

0, 1, 1, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 8, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 11

Offset: 1

Labos Elemer, May 29 2002

Erdős proved that there exist two constants c1, c2 > 0 such that c1 (n / log(n))^(1/2) < a(n) < c2 (n / log(n))^(1/2). - Carlo Sanna, May 28 2019

R. Heyman and R. Miraj proved that the cardinality of the set { floor(n/p) : p <= n, p prime } is same as the number of distinct exponents in the prime factorization of n!. - Md Rahil Miraj, Apr 05 2024

			n=7: 7! = 5040 = 2*2*2*2*3*3*5*7; three different exponents arise: 4, 2 and 1; a(7)=3.
n=7: { floor(7/p) : p <= 7, p prime } = {3,2,1}. So, its cardinality is 3. - _Md Rahil Miraj_, Apr 05 2024

David A. Corneth, Table of n, a(n) for n = 1..10000
Thomas Bloom, Problem 912, Erdős Problems. See Terence Tao's 31 August 2025 comment.
Paul Erdős, Miscellaneous problems in number theory, Proceedings of the Eleventh Manitoba Conference on Numerical Mathematics and Computing (Winnipeg, Man., 1981), Congressus Numerantium 34 (1982), 25-45.
Randell Heyman and Md Rahil Miraj, On some floor function sets, arXiv:2309.16072 [math.NT], 2023-2024.
Terence Tao, Erdős problem database, see no. 912.

Cf. A051903, A051904, A071625, A240751.
Cf. A000142, A001221, A001222, A011371, A022559, A076934, A115627, A135291.
Cf. A325272, A325273, A325276, A325508.

ffi[x_] := Flatten[FactorInteger[x]] lf[x_] := Length[FactorInteger[x]] ep[x_] := Table[Part[ffi[x], 2*w], {w, 1, lf[x]}] Table[Length[Union[ep[w! ]]], {w, 1, 100}]
Table[Length[Union[Last/@If[n==1,{},FactorInteger[n!]]]],{n,30}] (* Gus Wiseman, May 15 2019 *)

a(n) = #Set(factor(n!)[, 2]); \\ Michel Marcus, Sep 05 2017

a(n) = A071625(n!) = A323023(n!,3). - Gus Wiseman, May 15 2019

A115627 Irregular triangle read by rows: T(n,k) = multiplicity of prime(k) as a divisor of n!.

Original entry on oeis.org

1, 1, 1, 3, 1, 3, 1, 1, 4, 2, 1, 4, 2, 1, 1, 7, 2, 1, 1, 7, 4, 1, 1, 8, 4, 2, 1, 8, 4, 2, 1, 1, 10, 5, 2, 1, 1, 10, 5, 2, 1, 1, 1, 11, 5, 2, 2, 1, 1, 11, 6, 3, 2, 1, 1, 15, 6, 3, 2, 1, 1, 15, 6, 3, 2, 1, 1, 1, 16, 8, 3, 2, 1, 1, 1, 16, 8, 3, 2, 1, 1, 1, 1

Offset: 2

Franklin T. Adams-Watters, Jan 26 2006

The factorization of n! is n! = 2^T(n,1)*3^T(n,2)*...*p_(pi(n))^T(n,pi(n)) where p_k = k-th prime, pi(n) = A000720(n).
Nonzero terms of A085604; T(n,k) = A085604(n,k), k = 1..A000720(n). - Reinhard Zumkeller, Nov 01 2013
For n=2, 3, 4 and 5, all terms of the n-th row are odd. Are there other such rows? - Michel Marcus, Nov 11 2018
From Gus Wiseman, May 15 2019: (Start)
Differences between successive rows are A067255, so row n is the sum of the first n row-vectors of A067255 (padded with zeros on the right so that all n row-vectors have length A000720(n)). For example, the first 10 rows of A067255 are
  {}
  1
  0 1
  2 0
  0 0 1
  1 1 0
  0 0 0 1
  3 0 0 0
  0 2 0 0
  1 0 1 0
with column sums (8,4,2,1), which is row 10.
(End)
For all prime p > 7, 3*p > 2*nextprime(p), so for any n > 21 there will always be a prime p dividing n! with exponent 2 and there are no further rows with all entries odd. - Charlie Neder, Jun 03 2019

			From _Gus Wiseman_, May 09 2019: (Start)
Triangle begins:
   1
   1  1
   3  1
   3  1  1
   4  2  1
   4  2  1  1
   7  2  1  1
   7  4  1  1
   8  4  2  1
   8  4  2  1  1
  10  5  2  1  1
  10  5  2  1  1  1
  11  5  2  2  1  1
  11  6  3  2  1  1
  15  6  3  2  1  1
  15  6  3  2  1  1  1
  16  8  3  2  1  1  1
  16  8  3  2  1  1  1  1
  18  8  4  2  1  1  1  1
(End)
m such that 5^m||101!: floor(log(101)/log(5)) = 2 terms. floor(101/5) = 20. floor(20/5) = 4. So m = u_1 + u_2 = 20 + 4 = 24. - _David A. Corneth_, Jun 22 2014

T. D. Noe, Rows n = 2..300, flattened
H. T. Davis, Tables of the Mathematical Functions, Vols. 1 and 2, 2nd ed., 1963, Vol. 3 (with V. J. Fisher), 1962; Principia Press of Trinity Univ., San Antonio, TX. [Annotated scan of pages 204-208 of Volume 2.] See Table 2 on page 206.
Wenguang Zhai, On the prime power factorization of n!, Journal of Number Theory, Volume 129, Issue 8, August 2009, pages 1820-1836.
Index entries for sequences related to factorial numbers

Row lengths are A000720.
Row-sums are A022559.
Row-products are A135291.
Row maxima are A011371.
Columns include A011371, A054861, A027868, A054896, A090617, A064458, A090620.
Cf. A090622, A090623, A000142, A115628.
Cf. A085604, A141809.
Cf. A034876, A067255, A071626, A076934, A322583, A325272, A325273, A325276, A325508, A325509.

a115627 n k = a115627_tabf !! (n-2) !! (k-1)
a115627_row = map a100995 . a141809_row . a000142
a115627_tabf = map a115627_row [2..]
-- Reinhard Zumkeller, Nov 01 2013

A115627 := proc(n,k) local d,p; p := ithprime(k) ; n-add(d,d=convert(n,base,p)) ; %/(p-1) ; end proc: # R. J. Mathar, Oct 29 2010

Flatten[Table[Transpose[FactorInteger[n!]][[2]], {n, 2, 20}]] (* T. D. Noe, Apr 10 2012 *)
T[n_, k_] := Module[{p, jm}, p = Prime[k]; jm = Floor[Log[p, n]]; Sum[Floor[n/p^j], {j, 1, jm}]]; Table[Table[T[n, k], {k, 1, PrimePi[n]}], {n, 2, 20}] // Flatten (* Jean-François Alcover, Feb 23 2015 *)

a(n)=my(i=2);while(n-primepi(i)>1,n-=primepi(i);i++);p=prime(n-1);sum(j=1,log(i)\log(p),i\=p) \\ David A. Corneth, Jun 21 2014

T(n,k) = Sum_{i=1..inf} floor(n/(p_k)^i). (Although stated as an infinite sum, only finitely many terms are nonzero.)

T(n,k) = Sum_{i=1..floor(log(n)/log(p_k))} floor(u_i) where u_0 = n and u_(i+1) = floor((u_i)/p_k). - David A. Corneth, Jun 22 2014

A325617 Multinomial coefficient of the prime signature of n!.

Original entry on oeis.org

1, 1, 1, 2, 4, 20, 105, 840, 3960, 51480, 675675, 10810800, 139675536, 2793510720, 58663725120, 1799020903680, 26985313555200, 782574093100800, 25992639520848000, 857757104187984000, 30021498646579440000, 1563341744336692320000, 64179292662243158400000

Offset: 0

Gus Wiseman, May 12 2019

nonn

Number of permutations of the multiset of prime factors of n!.

			The a(5) = 20 permutations of {2,2,2,3,5}:
  (22235)  (32225)  (52223)
  (22253)  (32252)  (52232)
  (22325)  (32522)  (52322)
  (22352)  (35222)  (53222)
  (22523)
  (22532)
  (23225)
  (23252)
  (23522)
  (25223)
  (25232)
  (25322)

Wikipedia, Multinomial theorem

Cf. A000142, A008480, A011371, A022559, A076934, A108731, A115627, A318762, A325272, A325273, A325508, A325543, A325544, A325609.

Table[Multinomial@@Last/@FactorInteger[n!],{n,0,15}]

a(n) = A318762(A181819(n!)).

A325508 Product of primes indexed by the prime exponents of n!.

Original entry on oeis.org

1, 1, 2, 4, 10, 20, 42, 84, 204, 476, 798, 1596, 3828, 7656, 12276, 24180, 36660, 73320, 120840, 241680, 389424, 785680, 1294440, 2588880, 3848880, 7147920, 11264760, 15926040, 26057304, 52114608, 74421648, 148843296, 187159392, 340949280, 527531760, 926505360

Offset: 0

Gus Wiseman, May 08 2019

nonn

The prime indices of a(n) are the signature of n!, which is row n of A115627.

			We have 7! = 2^4 * 3^2 * 5^1 * 7^1, so a(7) = prime(4)*prime(2)*prime(1)*prime(1) = 84.
The sequence of terms together with their prime indices begins:
          1: {}
          1: {}
          2: {1}
          4: {1,1}
         10: {1,3}
         20: {1,1,3}
         42: {1,2,4}
         84: {1,1,2,4}
        204: {1,1,2,7}
        476: {1,1,4,7}
        798: {1,2,4,8}
       1596: {1,1,2,4,8}
       3828: {1,1,2,5,10}
       7656: {1,1,1,2,5,10}
      12276: {1,1,2,2,5,11}
      24180: {1,1,2,3,6,11}
      36660: {1,1,2,3,6,15}
      73320: {1,1,1,2,3,6,15}
     120840: {1,1,1,2,3,8,16}
     241680: {1,1,1,1,2,3,8,16}

Cf. A000142, A011371, A022559, A056171, A071626, A076934, A115627, A118914, A122111, A135291, A181819, A307035, A323014, A325272, A325273, A325276, A325509.

Table[Times@@Prime/@Last/@If[(n!)==1,{},FactorInteger[n!]],{n,0,30}]

a(n) = A181819(n!).
A001221(a(n)) = A071626(n).
A001222(a(n)) = A000720(n).
A056239(a(n)) = A022559(n).
A003963(a(n)) = A135291(n).
A061395(a(n)) = A011371(n).
A007814(a(n)) = A056171(n).
a(n) = A122111(A307035(n)). - Antti Karttunen, Nov 19 2019

A111701 Least integer obtained when n is divided by prime(1), then by prime(2), then by prime(3), ..., stopping as soon as one of the primes does not divide it. In particular, a(2n-1) = 2n-1.

Original entry on oeis.org

1, 1, 3, 2, 5, 1, 7, 4, 9, 5, 11, 2, 13, 7, 15, 8, 17, 3, 19, 10, 21, 11, 23, 4, 25, 13, 27, 14, 29, 1, 31, 16, 33, 17, 35, 6, 37, 19, 39, 20, 41, 7, 43, 22, 45, 23, 47, 8, 49, 25, 51, 26, 53, 9, 55, 28, 57, 29, 59, 2, 61, 31, 63, 32, 65, 11, 67, 34, 69, 35, 71, 12, 73, 37, 75, 38, 77

Offset: 1

Amarnath Murthy, Aug 21 2005

n divided by the largest primorial that divides it.

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

Cf. A002110, A053589, A076934, A114562.

f[n_] := Block[{m = n, k = 1}, While[ IntegerQ[ m/Prime[k]], m = m/Prime[k]; k++ ]; m]; Table[ f[n], {n, 77}] (* Robert G. Wilson v, Aug 25 2005 *)

a(n) = n/A053589(n). - Amiram Eldar, Mar 30 2021

Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Sum{k>=1} (1-1/prime(k))/(prime(k-1)#)^2 = 0.68986205818209290747..., where prime(k)# = A002110(k). - Amiram Eldar, Dec 25 2023

More terms from Robert G. Wilson v, Aug 25 2005

A325616 Triangle read by rows where T(n,k) is the number of length-k integer partitions of n into factorial numbers.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 0, 1, 0, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 1, 1, 2, 1, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 2, 1, 1, 1, 1, 1, 0, 0, 1, 0, 1, 1, 2, 2, 1

Offset: 0

Gus Wiseman, May 12 2019

			Triangle begins:
  1
  0 1
  0 1 1
  0 0 1 1
  0 0 1 1 1
  0 0 0 1 1 1
  0 1 0 1 1 1 1
  0 0 1 0 1 1 1 1
  0 0 1 1 1 1 1 1 1
  0 0 0 1 1 1 1 1 1 1
  0 0 0 1 1 2 1 1 1 1 1
  0 0 0 0 1 1 2 1 1 1 1 1
  0 0 1 0 1 1 2 2 1 1 1 1 1
  0 0 0 1 0 1 1 2 2 1 1 1 1 1
  0 0 0 1 1 1 1 2 2 2 1 1 1 1 1
  0 0 0 0 1 1 1 1 2 2 2 1 1 1 1 1
  0 0 0 0 1 1 2 1 2 2 2 2 1 1 1 1 1
  0 0 0 0 0 1 1 2 1 2 2 2 2 1 1 1 1 1
  0 0 0 1 0 1 1 2 2 2 2 2 2 2 1 1 1 1 1
  0 0 0 0 1 0 1 1 2 2 2 2 2 2 2 1 1 1 1 1
  0 0 0 0 1 1 1 1 2 2 3 2 2 2 2 2 1 1 1 1 1
Row n = 12 counts the following partitions:
  (66)
  (6222)
  (62211)
  (222222) (621111)
  (2222211) (6111111)
  (22221111)
  (222111111)
  (2211111111)
  (21111111111)
  (111111111111)

Row sums are A064986.
Cf. A008284.
Factorial numbers: A000142, A007489, A076934, A108731, A115944, A227157, A284605, A322583, A325509, A325617.
Reciprocal factorial sum: A325618, A325619, A325620, A325622.

Table[SeriesCoefficient[Product[1/(1-y*x^(i!)),{i,1,n}],{x,0,n},{y,0,k}],{n,0,15},{k,0,n}]

T(n,k) is the coefficient of x^n * y^k in the expansion of Product_{i > 0} 1/(1 - y * x^(i!)).

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

A093411 Divide n by the largest factorial that divides it and repeat until an odd number is reached, which will be the result; a(0) = 0.

Original entry on oeis.org

0, 1, 1, 3, 1, 5, 1, 7, 1, 9, 5, 11, 1, 13, 7, 15, 1, 17, 3, 19, 5, 21, 11, 23, 1, 25, 13, 27, 7, 29, 5, 31, 1, 33, 17, 35, 1, 37, 19, 39, 5, 41, 7, 43, 11, 45, 23, 47, 1, 49, 25, 51, 13, 53, 9, 55, 7, 57, 29, 59, 5, 61, 31, 63, 1, 65, 11, 67, 17, 69, 35, 71, 3, 73, 37, 75, 19, 77, 13, 79

Offset: 0

Amarnath Murthy, Mar 30 2004

a(n) is odd for all positive n>0; a(n) = n iff n is odd.

			a(18) = 3, 18/6 = 3. though 18/2 = 9.

Antti Karttunen, Table of n, a(n) for n = 0..40319
Index entries for sequences related to factorial numbers

For bisection see A109375, for positions of ones, A344181.
Cf. A000142, A076934, A329379.
Cf. also A328478.

f[n_] := Block[{m = n, k = n}, While[k > 1, While[ IntegerQ[m/k! ], m = m/k! ]; k-- ]; m]; Table[ f[n], {n, 0, 79}] (* Robert G. Wilson v *)

A076934(n) = for(k=2, oo , if(n%k, return(n), n /= k));
A093411(n) = if(!n,n, if(n%2, n, A093411(A076934(n)))); \\ Antti Karttunen, May 18 2021

From Antti Karttunen, May 18 2021: (Start)
a(0) = 0, a(2n+1) = 2n+1, a(2n) = a(A076934(2n)).
a(n) = n / A329379(n).
(End)

Edited, corrected and extended by Robert G. Wilson v, Aug 25 2005

Definition further clarified by Antti Karttunen, May 18 2021

A325543 Width (number of leaves) of the rooted tree with Matula-Goebel number n!.

Original entry on oeis.org

1, 1, 1, 2, 4, 5, 7, 9, 12, 14, 16, 17, 20, 22, 25, 27, 31, 33, 36, 39, 42, 45, 47, 49, 53, 55, 58, 61, 65, 67, 70, 71, 76, 78, 81, 84, 88, 91, 95, 98, 102, 104, 108, 111, 114, 117, 120, 122, 127, 131, 134, 137, 141, 145, 149, 151, 156, 160, 163, 165, 169, 172

Offset: 0

Gus Wiseman, May 09 2019

nonn

Also the multiplicity of q(1) in the factorization of n! into factors q(i) = prime(i)/i. For example, the factorization of 7! is q(1)^9 * q(2)^3 * q(3) * q(4), so a(7) = 9.

			Matula-Goebel trees of the first 9 factorial numbers are:
  0!: o
  1!: o
  2!: (o)
  3!: (o(o))
  4!: (ooo(o))
  5!: (ooo(o)((o)))
  6!: (oooo(o)(o)((o)))
  7!: (oooo(o)(o)((o))(oo))
  8!: (ooooooo(o)(o)((o))(oo))
The number of leaves is the number of o's, which are (1, 1, 1, 2, 4, 5, 7, 9, 12, ...), as required.

Cf. A000081, A001222, A056239, A324922, A324923, A324924.
Matula-Goebel numbers: A007097, A061775, A109082, A109129, A196050, A317713.
Factorial numbers: A000142, A011371, A022559, A071626, A076934, A115627, A325272, A325273, A325276, A325508, A325544.

mglv[n_]:=If[n==1,1,Total[Cases[FactorInteger[n],{p_,k_}:>mglv[PrimePi[p]]*k]]];
Table[mglv[n!],{n,0,100}]

For n > 1, a(n) = - 1 + Sum_{k = 1..n} A109129(k).

A325544 Number of nodes in the rooted tree with Matula-Goebel number n!.

Original entry on oeis.org

1, 1, 2, 4, 6, 9, 12, 15, 18, 22, 26, 30, 34, 38, 42, 47, 51, 55, 60, 64, 69, 74, 79, 84, 89, 95, 100, 106, 111, 116, 122, 127, 132, 138, 143, 149, 155, 160, 165, 171, 177, 182, 188, 193, 199, 206, 212, 218, 224, 230, 237, 243, 249, 254, 261, 268, 274, 280

Offset: 0

Gus Wiseman, May 09 2019

nonn

Also one plus the number of factors in the factorization of n! into factors q(i) = prime(i)/i. For example, the q-factorization of 7! is 7! = q(1)^9 * q(2)^3 * q(3) * q(4), with 14 = a(7) - 1 factors.

			Matula-Goebel trees of the first 9 factorial number are:
  0!: o
  1!: o
  2!: (o)
  3!: (o(o))
  4!: (ooo(o))
  5!: (ooo(o)((o)))
  6!: (oooo(o)(o)((o)))
  7!: (oooo(o)(o)((o))(oo))
  8!: (ooooooo(o)(o)((o))(oo))
The number of nodes is the number of o's plus the number of brackets, giving {1,1,2,4,6,9,12,15,18}, as required.

Cf. A000081, A001222, A056239, A324922, A324923, A324924.
Matula-Goebel numbers: A007097, A061775, A109082, A109129, A196050, A317713.
Factorial numbers: A000142, A011371, A022559, A071626, A076934, A115627, A325272, A325273, A325276, A325508, A325543.

mgwt[n_]:=If[n==1,1,1+Total[Cases[FactorInteger[n],{p_,k_}:>mgwt[PrimePi[p]]*k]]];
Table[mgwt[n!],{n,0,100}]

For n > 1, a(n) = 1 - n + Sum_{k = 1..n} A061775(k).

cp's OEIS Frontend

A071626 Number of distinct exponents in the prime factorization of n!.

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A115627 Irregular triangle read by rows: T(n,k) = multiplicity of prime(k) as a divisor of n!.

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A325617 Multinomial coefficient of the prime signature of n!.

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A325508 Product of primes indexed by the prime exponents of n!.

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A111701 Least integer obtained when n is divided by prime(1), then by prime(2), then by prime(3), ..., stopping as soon as one of the primes does not divide it. In particular, a(2n-1) = 2n-1.

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A325616 Triangle read by rows where T(n,k) is the number of length-k integer partitions of n into factorial numbers.

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

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A093411 Divide n by the largest factorial that divides it and repeat until an odd number is reached, which will be the result; a(0) = 0.

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