|
%I #19 Jan 04 2019 16:25:23
%S 2,2,3,8,3,8,3,5,16,9,5,16,9,5,7,128,9,5,7,128,81,5,7,256,81,25,7,256,
%T 81,25,7,11,1024,243,25,7,11,1024,243,25,7,11,13,2048,243,25,49,11,13,
%U 2048,729,125,49,11,13,32768,729,125,49,11,13,32768,729,125,49,11,13,17,65536
%N Triangle of the powers of the prime factorization of n! in row n.
%C Row n contains pi(n) = A000720(n) terms. The exponents are in A115627.
%C The first column contains the maximum power of 2 dividing n!, the second column the maximum power of 3 dividing n! etc.
%H Alois P. Heinz, <a href="/A176383/b176383.txt">Rows n = 2..300, flattened</a>
%e The irregular tables starts with n=2:
%e 2; # =2! = 2
%e 2*3; # =3! = 6
%e 8*3; # =4! = 24
%e 8*3*5; # =5! = 120
%e 16*9*5; # =6!
%e 16*9*5*7; # =7!
%e 128*9*5*7; # =8!
%e 128*81*5*7;
%e 256*81*25*7;
%p with(numtheory):
%p b:= proc(n) option remember; `if`(n=1, 1, b(n-1)+
%p add(i[2]*x^pi(i[1]), i=ifactors(n)[2]))
%p end:
%p T:= n->(p->seq(ithprime(i)^coeff(p, x, i), i=1..pi(n)))(b(n)):
%p seq(T(n), n=2..20); # _Alois P. Heinz_, Jun 22 2014
%t T[n_] := List @@ Power @@@ FactorInteger[n!];
%t Array[T, 20, 2] // Flatten (* _Jean-François Alcover_, Mar 27 2017 *)
%t Rest[Flatten[Table[#[[1]]^#[[2]]&/@FactorInteger[n!],{n,20}]]] (* _Harvey P. Dale_, Jan 04 2019 *)
%o (PARI) a(n)=my(i=2);while(n-primepi(i)>1,n-=primepi(i);i++);p=prime(n-1);p^sum(j=1,log(i)\log(p),i\=p) \\ _David A. Corneth_, Jun 21 2014
%Y Cf. A000142, A000720.
%K nonn,tabf
%O 2,1
%A _Vladimir Shevelev_, Apr 16 2010
%E Arbitrarily defined first 2 terms removed by _R. J. Mathar_, Apr 23 2010
|
