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A176383 Triangle of the powers of the prime factorization of n! in row n.

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