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

2,1

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.

Links

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

Example

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;

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 *)

Prog

(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

Crossrefs

Cf. A000142, A000720.

Sequence in context: A204197 A238654 A220553 * A100551 A267667 A210190

Adjacent sequences: A176380 A176381 A176382 * A176384 A176385 A176386

Keyword

nonn,tabf

Author

Vladimir Shevelev, Apr 16 2010

Extensions

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

Status

approved