A307035 a(n) is the unique integer k such that A108951(k) = n!.

1, 1, 2, 3, 12, 20, 60, 84, 672, 1512, 5040, 7920, 47520, 56160, 157248, 393120, 6289920, 8225280, 37013760, 41368320, 275788800, 579156480, 1820206080, 2203407360, 26440888320, 73446912000, 173601792000, 585906048000, 3281073868800, 4137006182400, 20685030912000

Offset: 0

Author

Allan C. Wechsler, Mar 20 2019

Keywords

• nonn
• easy

Comments

For all n, n! = A108951(k) for some unique k. This sequence gives that k for each n. In some sense this sequence tells how to factor factorials into primorials.

Represent n! as a product of primorials p#. Then replace each primorial with its base prime to calculate a(n).

Examples

			Represent 7! as a product of primorials:
7! = 2^4 * 3^2 * 5 * 7 = (2#)^2 * 3# * 7#
Replace primorials by primes:
2^2 * 3 * 7 = 84.
So a(7) = 84.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1911 (terms n = 1..1000 from Vaclav Kotesovec)

Crossrefs

Cf. A000142, A108951, A025487, A122111, A319626, A319627, A325508, A329900.

Programs

Maple

f:= proc(n) option remember; `if`(n<2, 0, f(n-1)+add(
      i[2]*x^numtheory[pi](i[1]), i=ifactors(n)[2]))
    end:
a:= proc(n) local d, p, r; p, r:= f(n), 1;
      do d:= degree(p); if d<1 then break fi;
         p, r:= p-add(x^i, i=1..d), ithprime(d)*r
      od: r
    end:
seq(a(n), n=0..35);  # Alois P. Heinz, Mar 21 2019

Mathematica

q[n_] := Apply[Times, Table[Prime[i], {i, 1, PrimePi[n]}]]; Flatten[{1, 1, Table[val = 1; fak = n!; Do[If[PrimeQ[k], Do[If[Divisible[fak, q[k]], val = val*k; fak = fak/q[k]], {j, 1, n}]], {k, n, 2, -1}]; val, {n, 2, 30}]}] (* Vaclav Kotesovec, Mar 21 2019 *)

PARI

g(n) = my(f=factor(n)); prod(k=1, #f~, my(p=f[k, 1]); (p/if(p>2, precprime(p-1), 1))^f[k, 2]); \\ A319626/A319627
a(n) = prod(k=1, n, g(k)); \\ Daniel Suteu, Mar 21 2019

PARI

A307035(n) = { my(m=1, pp=1); n=n!; while(1, forprime(p=2, ,if(n%p, if(2==p, return(m), break), n /= p; pp = p)); m *= pp); }; \\ Antti Karttunen, Dec 29 2019

Formula

a(0) = 1, a(n) = a(n-1) * (A319626(n) / A319627(n)), for n > 0. - Daniel Suteu, Mar 21 2019

a(n) = n! / Product_{k=1..n} A064989(k). - Vaclav Kotesovec, Mar 21 2019

a(n) = A122111(A325508(n)) = A319626(A000142(n)) = A329900(A000142(n)). - Antti Karttunen, Nov 19 & Dec 29 2019

Extensions

a(12)-a(13) from Michel Marcus, Mar 21 2019

a(14)-a(15) from Vaclav Kotesovec, Mar 21 2019

a(0), a(16)-a(30) from Alois P. Heinz, Mar 21 2019