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A115627 Irregular triangle read by rows: T(n,k) = multiplicity of prime(k) as a divisor of n!. 16
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

2,4

COMMENTS

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

LINKS

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.

FORMULA

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

EXAMPLE

Rows start:

1;

1,1;

3,1;

3,1,1;

4,2,1;

4,2,1,1;

7,2,1,1;

7,4,1,1;

...

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, 22 Jun 2014

MAPLE

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

MATHEMATICA

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-Fran├žois Alcover, Feb 23 2015 *)

PROG

(Haskell)

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

(PARI) 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, 21 Jun 2014

CROSSREFS

Cf. A090622, A090623, A000142, A115628.

Row lengths are A000720.

Columns include A011371, A054861, A027868, A054896, A090617, A064458, A090620.

Cf. also A085604, A141809.

Sequence in context: A225212 A091088 A249781 * A128218 A010283 A134699

Adjacent sequences:  A115624 A115625 A115626 * A115628 A115629 A115630

KEYWORD

nonn,tabf

AUTHOR

Franklin T. Adams-Watters, Jan 26 2006

STATUS

approved

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Last modified March 25 01:17 EDT 2019. Contains 321450 sequences. (Running on oeis4.)