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%I #88 Mar 03 2024 09:30:37
%S 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,
%T 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,
%U 1,1,16,8,3,2,1,1,1,16,8,3,2,1,1,1,1
%N Irregular triangle read by rows: T(n,k) = multiplicity of prime(k) as a divisor of n!.
%C 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).
%C Nonzero terms of A085604; T(n,k) = A085604(n,k), k = 1..A000720(n). - _Reinhard Zumkeller_, Nov 01 2013
%C 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
%C From _Gus Wiseman_, May 15 2019: (Start)
%C 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
%C {}
%C 1
%C 0 1
%C 2 0
%C 0 0 1
%C 1 1 0
%C 0 0 0 1
%C 3 0 0 0
%C 0 2 0 0
%C 1 0 1 0
%C with column sums (8,4,2,1), which is row 10.
%C (End)
%C 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
%H T. D. Noe, <a href="/A115627/b115627.txt">Rows n = 2..300, flattened</a>
%H H. T. Davis, <a href="/A002443/a002443.pdf">Tables of the Mathematical Functions</a>, 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.
%H Wenguang Zhai, <a href="http://dx.doi.org/10.1016/j.jnt.2009.02.016">On the prime power factorization of n!</a>, Journal of Number Theory, Volume 129, Issue 8, August 2009, pages 1820-1836.
%H <a href="/index/Fa#factorial">Index entries for sequences related to factorial numbers</a>
%F T(n,k) = Sum_{i=1..inf} floor(n/(p_k)^i). (Although stated as an infinite sum, only finitely many terms are nonzero.)
%F 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
%e From _Gus Wiseman_, May 09 2019: (Start)
%e Triangle begins:
%e 1
%e 1 1
%e 3 1
%e 3 1 1
%e 4 2 1
%e 4 2 1 1
%e 7 2 1 1
%e 7 4 1 1
%e 8 4 2 1
%e 8 4 2 1 1
%e 10 5 2 1 1
%e 10 5 2 1 1 1
%e 11 5 2 2 1 1
%e 11 6 3 2 1 1
%e 15 6 3 2 1 1
%e 15 6 3 2 1 1 1
%e 16 8 3 2 1 1 1
%e 16 8 3 2 1 1 1 1
%e 18 8 4 2 1 1 1 1
%e (End)
%e 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
%p 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
%t Flatten[Table[Transpose[FactorInteger[n!]][[2]], {n, 2, 20}]] (* _T. D. Noe_, Apr 10 2012 *)
%t 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 *)
%o (Haskell)
%o a115627 n k = a115627_tabf !! (n-2) !! (k-1)
%o a115627_row = map a100995 . a141809_row . a000142
%o a115627_tabf = map a115627_row [2..]
%o -- _Reinhard Zumkeller_, Nov 01 2013
%o (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_, Jun 21 2014
%Y Row lengths are A000720.
%Y Row-sums are A022559.
%Y Row-products are A135291.
%Y Row maxima are A011371.
%Y Columns include A011371, A054861, A027868, A054896, A090617, A064458, A090620.
%Y Cf. A090622, A090623, A000142, A115628.
%Y Cf. A085604, A141809.
%Y Cf. A034876, A067255, A071626, A076934, A322583, A325272, A325273, A325276, A325508, A325509.
%K nonn,tabf
%O 2,4
%A _Franklin T. Adams-Watters_, Jan 26 2006
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