A115627 -id:A115627 - cp's OEIS frontend

A034876 Number of ways to write n! as a product of smaller factorials each greater than 1.

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

Offset: 1

Erich Friedman

By definition, a(n) > 0 if and only if n is a member of A034878. If n > 2, then a(n!) > max(a(n), a(n!-1)), as (n!)! = n!*(n!-1)!. Similarly, a(A001013(n)) > 0 for n > 2. Clearly a(n)=0 if n is a prime A000040. So a(n+1)=1 if n=2^p-1 is a Mersenne prime A000668, as (n+1)!=(2!)^p*n! and n is prime. - Jonathan Sondow, Dec 15 2004
From Antti Karttunen, Dec 25 2018: (Start)
If n! = a! * x! * y! * ... * z!, with a > x >= y >= z, then A006530(n!) = A006530(a!) > A006530(x!). This follows because all rows in A115627 end with 1, that is, because all factorials >= 2 are in A102750.
If all the two-term solutions are of the form n! = a! * x! = b! * y! = ... = c! * z! (that is, all are products of two factorials larger than one), with a > x, b > y, ..., c > z, then a(n) = (a(x)+1 + a(y)+1 + ... + a(z)+1).
Values 0..5 occur for the first time at n = 1, 4, 10, 576, 13824, 69120.
In range 1..69120 differs from A322583 only at positions n = 1, 2, 9, 10 and 16.
(End)

			a(10) = 2 because 10! = 3! * 5! * 7! = 6! * 7! are the only two ways to write 10! as a product of smaller factorials > 1.
From _Antti Karttunen_, Dec 25 2018: (Start)
a(8) = 1 because 8! = 7! * (2!)^3.
a(9) = 1 because 9! = 7! * 3! * 3! * 2!.
a(16) = 2 because 16! = 15! * (2!)^4 = 14! * 5! * 2!.
a(144) = 2 because 144! = 143! * 4! * 3! = 143! * 3! * 3! * 2! * 2!.
a(576) = 3 because 576! = 575! * 4! * 4! = 575! * 4! * 3! * 2! * 2! = 575! * 3! * 3! * 2! * 2! * 2! * 2!.
a(720) = 2 because 720! = 719! * 6! = 719! * 5! * 3!.
a(3456) = 3 because 3456! = 3455! * 4! * 4! * 3! = 3455! * 4! * 3! * 3! * 2! * 2! = 3455! * 3! * 3! * 3! * 2! * 2! * 2! * 2!.
(End)

R. K. Guy, Unsolved Problems in Number Theory, B23.

Antti Karttunen, Table of n, a(n) for n = 1..69120
Eric Weisstein's World of Mathematics, Factorial Products
Index entries for sequences related to factorial numbers

Cf. A034878, A001013, A075082, A085604, A115627, A322583.

A034876aux(n, m, p) = if(1==n, 1, my(s=0); forstep(i=m, p, -1, my(f=i!); if(!(n%f), s += A034876aux(n/f, i, 2))); (s));
A034876(n) = if(1==n,0,A034876aux(n!, n-1, precprime(n))); \\ (Slow) - Antti Karttunen, Dec 24 2018

A322583aux(n, m) = if(1==n, 1, my(s=0); for(i=2, oo, my(f=i!); if(f>m, return(s)); if(!(n%f), s += A322583aux(n/f, f))));
memoA322583 = Map();
A322583(n) = { my(c); if(mapisdefined(memoA322583,n,&c), c, c = A322583aux(n,n); mapput(memoA322583,n,c); (c)); };
A034876aux(n, m, p) = if(1==n, 1, my(s=0); forstep(i=m, p, -1, my(f=i!); s += A322583(n/f)); (s));
A034876(n) = if(1==n, 0, A034876aux(n!, n-1, precprime(n))); \\ Antti Karttunen, Dec 25 2018

a(1) = 0; for n > 1, a(n) = Sum_{x=A007917(n)..n-1} A322583(n!/x!) when n is a composite, and a(n) = 0 when n is a prime. - Antti Karttunen, Dec 25 2018

Corrected by Jonathan Sondow, Dec 18 2004

A249344 A(n,k) = exponent of the largest power of n-th prime which divides k, square array read by antidiagonals.

Original entry on oeis.org

0, 1, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Antti Karttunen, Oct 28 2014

Square array A(n,k), where n = row, k = column, read by antidiagonals: A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), ... (transpose of array A060175).
A(n,k) is the (p_n)-adic valuation of k, where p_n is the n-th prime, A000040(n).
Each row is effectively a ruler function, s, with s(1) = 0. - Peter Munn, Apr 30 2022

			The top-left corner of the array:
  0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 4, ...
  0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 0, 1, 0, 0, 1, 0, ...
  0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, ...
  0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, ...
  0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, ...
  0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, ...
  ...
A(1,8) = 3, because 2^3 is the largest power of 2 (= p_1 = A000040(1)) that divides 8.
a(2,9) = 2, because 3^2 is the largest power of 3 (= p_2) that divides 9.
a(3,15) = 1, because 5^1 is the largest power of 5 (= p_3) that divides 15.

Transpose: A060175.
Row 1: A007814.
Row 2: A007949.
Row 3: A112765.
Row 4: A214411.
Cf. also A000040, A002260, A004736, A085604, A090622, A115627, A249421.
Completely additive sequences where more than one prime is mapped to 1, all other primes to 0: A065339, A083025, A087436, A169611.
Ruler functions, s, with s(1) = 0 that are not rows here: A122840, A122841, A235127, A244413.

A[n_, k_] := IntegerExponent[k, Prime[n]]; Table[A[k, n - k + 1], {n, 1, 15}, {k, 1, n}] // Flatten (* Amiram Eldar, Oct 01 2023 *)

a(n, k) = valuation(k, prime(n)); \\ Michel Marcus, Jun 24 2017

from sympy import prime
def a(n, k):
    p=prime(n)
    i=z=0
    while p**i<=k:
        if k%(p**i)==0: z=i
        i+=1
    return z
for n in range(1, 10): print([a(k, n - k + 1) for k in range(1, n + 1)]) # Indranil Ghosh, Jun 24 2017

(define (A249344 n) (A249344bi (A002260 n) (A004736 n)))
(define (A249344bi row col) (let ((p (A000040 row))) (let loop ((n col) (i 0)) (cond ((not (zero? (modulo n p))) i) (else (loop (/ n p) (+ i 1)))))))

Row n, as a sequence, is completely additive with A(n, prime(n)) = 1, A(n, prime(m)) = 0 for m <> n. - Peter Munn, Apr 30 2022

Sum_{k=1..m} A(n,k) ~ (1/(prime(n)-1)) * m. - Amiram Eldar, Oct 01 2023

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]],#]!={}&]

A085604 T(n,k) = highest power of prime(k) dividing n!, read by rows.

Original entry on oeis.org

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

Offset: 1

Reinhard Zumkeller, Jul 07 2003

T(n,1) = A011371(n); T(n,2) = A054861(n) for n>1;
T(n,k) = number of occurrences of prime(k) as factor in numbers <= n (with repetitions);
Sum{T(n,k): 1<=k<=n} = A022559(n);
T(n, A000720(n)) = 1; T(n,k) = 0, A000720(n)T(n,k) = A115627(n,k) for n > 1 and k=1..A000720(n). - Reinhard Zumkeller, Nov 01 2013

			0;
1,0;
1,1,0;
3,1,0,0;
3,1,1,0,0;
4,2,1,0,0,0;
4,2,1,1,0,0,0;
7,2,1,1,0,0,0,0;
7,4,1,1,0,0,0,0,0;
8,4,2,1,0,0,0,0,0,0;

Reinhard Zumkeller, Rows n = 1..125 of triangle, flattened

Cf. A141809, A115627, A000142.

a085604 n k = a085604_tabl !! (n-2) !! (k-1)
a085604_row 1 = [0]
a085604_row n = a115627_row n ++ (take $ a062298 $ fromIntegral n) [0,0..]
a085604_tabl = map a085604_row [1..]
-- Reinhard Zumkeller, Nov 01 2013

T[n_, k_] := Module[{p = Prime[k], jm}, jm = Floor[Log[p, n]]; Sum[Quotient[n, p^j], {j, 1, jm}]];
Table[T[n, k], {n, 1, 14}, {k, 1, n}] // Flatten (* Jean-François Alcover, Sep 19 2021 *)

A177349 Primes p for which no m! has a prime power factorization of the form 2^p...p^1*...

Original entry on oeis.org

2, 5, 13, 17, 29, 37, 43, 59, 61, 83, 103, 107, 139, 151, 157, 163, 167, 179, 199, 211, 223, 227, 233, 241, 251, 257, 269, 283, 307, 313, 317, 331, 347, 373, 379, 409, 419, 433, 443, 457, 503, 509, 523, 541, 547, 563, 569, 571, 587, 601, 607, 617, 619, 643

Offset: 1

Vladimir Shevelev, May 07 2010

nonn

One can prove that for p>=5 the number of such m is 0 or 2.

			p=109 is not in the sequence because for m=112 we have 112! = 2^109*3^54*...*109 which is of the form 2^p*...*p*.. [_R. J. Mathar_, Oct 29 2010]

Amiram Eldar, Table of n, a(n) for n = 1..10000
Vladimir Shevelev, Compact integers and factorials, Acta Arithmetica 126 (2007) 195-236.

Cf. A000142.

isA177349 := proc(p) if isprime(p) then pid := numtheory[pi](p) ; for m from 1 do h2 := A115627(m,1) ; if h2 > p then return true; elif h2 = p then if A115627(m,pid) = 1 then return false; end if; end if; end do; else false; fi ; end proc:
for i from 1 to 120 do p := ithprime(i) ; if isA177349(p) then printf("%d,",p); end if; end do: # R. J. Mathar, Oct 29 2010

c[n_, p_] := Sum[IntegerExponent[k, p], {k, 2, n}]; m = 120; v = Table[0, {m}]; s = 0; n = 2; While[s <= Prime[m], s += IntegerExponent[n, 2]; If[ PrimeQ[s] && (i = PrimePi[s]) <= m && c[n, s] == 1, v[[i]] = 1]; n++]; t = Prime /@ (Position[v, ?(# == 0 &)] // Flatten); t (* _Amiram Eldar, Sep 13 2019 *)

2 added, 109 replaced by 107, sequence extended beyond 199 by R. J. Mathar, Oct 29 2010

A177458 The number of positive integers m for which the exponents of prime(n) and prime(n+1) in the prime power factorization of m! are both powers of 2.

Original entry on oeis.org

9, 22, 23, 22, 42, 37, 40, 90, 63, 96, 147, 120, 111, 134, 237, 166, 219, 304, 214, 279, 254, 252, 369, 484, 399, 520, 429, 270, 519, 481, 709, 426, 793, 581, 611, 734, 661, 691, 1003, 615, 1087, 914, 1129, 647, 707, 1094, 1339, 1130, 1032, 1423, 915, 1140

Offset: 3

Vladimir Shevelev, May 09 2010, May 10 2010

nonn

This gives the number of rows in A115627 for which the n-th and (n+1)st column are both in {1,2,4,8,16,..}.
For n=2 the corresponding value is not known and >=25; moreover, we do not know if this value is finite.
A more general result concerning the cases for non-adjacent primes and a finite search interval for the values of m is in the 2007 publication.

			For n=3, the 9 values of m are 7, 8, 9, 10, 11, 12, 13, 14, and 20.
m=6, for example, is not counted because 6!=2^4*3^2*5 does not contain prime(4)=7.
m=15, for example, is not counted because 15!=2^11*3^6*5^3*7^2*11*13 contains a third power of prime(3)=5.

Vladimir Shevelev, Compact integers and factorials, Acta Arithmetica 126 (2007), no. 3, 195-236.

Cf. A000142, A177355, A177349, A177378, A177436.

tp[n_] := Flatten[Position[FoldList[Plus, 0, IntegerExponent[Range[100000], n]], ?(IntegerQ[Log[2, #]] &)]]; Table[s = Intersection[tp[Prime[n]], tp[Prime[n + 1]]] - 1; Length[s], {n, 3, 60}] (* _T. D. Noe, Apr 10 2012 *)

Edited, example and relation to A115627 added, terms after 120 added by R. J. Mathar, Oct 29 2010

Extended by T. D. Noe, Apr 10 2012

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).

A336497 Numbers that cannot be written as a product of superfactorials A000178.

Original entry on oeis.org

3, 5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76

Offset: 1

Gus Wiseman, Aug 03 2020

nonn

First differs from A336426 in having 360.

			The sequence of terms together with their prime indices begins:
     3: {2}        22: {1,5}        39: {2,6}
     5: {3}        23: {9}          40: {1,1,1,3}
     6: {1,2}      25: {3,3}        41: {13}
     7: {4}        26: {1,6}        42: {1,2,4}
     9: {2,2}      27: {2,2,2}      43: {14}
    10: {1,3}      28: {1,1,4}      44: {1,1,5}
    11: {5}        29: {10}         45: {2,2,3}
    13: {6}        30: {1,2,3}      46: {1,9}
    14: {1,4}      31: {11}         47: {15}
    15: {2,3}      33: {2,5}        49: {4,4}
    17: {7}        34: {1,7}        50: {1,3,3}
    18: {1,2,2}    35: {3,4}        51: {2,7}
    19: {8}        36: {1,1,2,2}    52: {1,1,6}
    20: {1,1,3}    37: {12}         53: {16}
    21: {2,4}      38: {1,8}        54: {1,2,2,2}

A093373 is the version for factorials, with complement A001013.
A336426 is the version for superprimorials, with complement A181818.
A336496 is the complement.
A000178 lists superfactorials.
A001055 counts factorizations.
A006939 lists superprimorials or Chernoff numbers.
A049711 is the minimum prime multiplicity in A000178(n).
A174605 is the maximum prime multiplicity in A000178(n).
A303279 counts prime factors (with multiplicity) of superprimorials.
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[100],facsusing[Rest[Array[supfac,30]],#]=={}&]

A266743 Irregular triangle T(n,k) read by rows: see Comments for definition.

Original entry on oeis.org

1, 1, 1, 2, 3, 1, 1, 2, 1, 6, 15, 10, 1, 2, 6, 5, 1, 12, 42, 42, 14, 1, 3, 12, 14, 7, 1, 10, 45, 60, 42, 10, 1, 2, 10, 15, 14, 5, 1, 12, 66, 110, 132, 66, 22, 1, 2, 12, 22, 33, 22, 11, 1, 420, 2730, 5460, 10010, 8580, 6006, 910, 1

Offset: 1

N. J. A. Sloane, Jan 08 2016

Let p_i denote the i-th prime, let pi(n) = A000720(n), and let N! = Product_{i = 1..pi(N)} (p_i)^U(N,i) be the prime factorization of N!, where U(N,i) = A115627(N,i).
Let V(n,i) = floor(n/(prime(i)-1)) = A266742(n,i).
The present triangle is defined by T(n,k) =
Product_{i} (p_i)^V(n,i) / ( Product_{j} (p_j)^V(k,j) * Product_{r} (p_r)^U(n-k+1,r) ).

			Triangle begins:
    1;
    1,    1;
    2,    3,    1;
    1,    2,    1;
    6,   15,   10,     1;
    2,    6,    5,     1;
   12,   42,   42,    14,    1;
    3,   12,   14,     7,    1;
   10,   45,   60,    42,   10,    1;
    2,   10,   15,    14,    5,    1;
   12,   66,  110,   132,   66,   22,   1;
    2,   12,   22,    33,   22,   11,   1;
  420, 2730, 5460, 10010, 8580, 6006, 910, 1;
  ...

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 3 on page 207.

Cf. A000720, A002443, A002444, A115627, A266742.

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

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A177349 Primes p for which no m! has a prime power factorization of the form 2^p...p^1*...