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

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, 81, 25, 7, 11, 1024, 243, 25, 7, 11, 1024, 243, 25, 7, 11, 13, 2048, 243, 25, 49, 11, 13, 2048, 729, 125, 49, 11, 13, 32768, 729, 125, 49, 11, 13, 32768, 729, 125, 49, 11, 13, 17, 65536

Offset: 2

Author

Vladimir Shevelev, Apr 16 2010

Keywords

• nonn
• tabf

Comments

Row n contains pi(n) = A000720(n) terms. The exponents are in A115627.

The first column contains the maximum power of 2 dividing n!, the second column the maximum power of 3 dividing n! etc.

Examples

			The irregular tables starts with n=2:
2; # =2! = 2
2*3; # =3! = 6
8*3; # =4! = 24
8*3*5; # =5! = 120
16*9*5; # =6!
16*9*5*7; # =7!
128*9*5*7; # =8!
128*81*5*7;
256*81*25*7;
		

Links

• Alois P. Heinz, Rows n = 2..300, flattened

Crossrefs

Cf. A000142, A000720.

Programs

Maple

with(numtheory):
b:= proc(n) option remember; `if`(n=1, 1, b(n-1)+
      add(i[2]*x^pi(i[1]), i=ifactors(n)[2]))
    end:
T:= n->(p->seq(ithprime(i)^coeff(p, x, i), i=1..pi(n)))(b(n)):
seq(T(n), n=2..20);  # Alois P. Heinz, Jun 22 2014

Mathematica

T[n_] := List @@ Power @@@ FactorInteger[n!];
Array[T, 20, 2] // Flatten (* Jean-François Alcover, Mar 27 2017 *)
Rest[Flatten[Table[#[[1]]^#[[2]]&/@FactorInteger[n!],{n,20}]]] (* Harvey P. Dale, Jan 04 2019 *)

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

Extensions

Arbitrarily defined first 2 terms removed by R. J. Mathar, Apr 23 2010