A177333 Smallest factor in the factorization of n! over distinct terms of A050376.

2, 2, 2, 2, 5, 5, 2, 2, 7, 7, 3, 3, 2, 2, 2, 2, 5, 5, 4, 3, 2, 2, 4, 4, 2, 2, 2, 2, 4, 4, 2, 2, 3, 3, 3, 3, 2, 2, 4, 4, 2, 2, 2, 2, 3, 3, 4, 4, 2, 2, 2, 2, 4, 4, 2, 2, 3, 3, 7, 7, 2, 2, 2, 2, 3, 3, 3, 4, 2, 2, 4, 4, 2, 2, 2, 2, 4, 4, 4, 4, 2, 2, 2, 2, 3, 4, 2, 2, 4, 4, 5, 3, 2, 2, 4, 4, 2, 2, 2, 2, 3, 3, 2, 2, 4

Offset: 2

Author

Vladimir Shevelev, May 06 2010

Keywords

• nonn

Examples

			The factorization of 10! = 3628800 is 2^8*3^4*5^2*7^1, where 2^8 > 3^4 > 5^2 > 7, so a(10)=7 is the smallest of these 4 factors.
		

References

• V. S. Shevelev, Multiplicative functions in the Fermi-Dirac arithmetic, Izvestia Vuzov of the North-Caucasus region, Nature sciences 4 (1996), 28-43 [Russian].

Links

• Amiram Eldar, Table of n, a(n) for n = 2..1000
• S. Litsyn and V. S. Shevelev, On factorization of integers with restrictions on the exponent, INTEGERS: Electronic Journal of Combinatorial Number Theory, 7 (2007), #A33, 1-36.

Crossrefs

Cf. A050376, A177329, A055460, A176525, A001358, A176472, A176509, A064380, A050292.

Programs

Maple

A177333 := proc(n) local a,p,pow2 ; a := n! ; for p in ifactors(n!)[2] do pow2 := convert( op(2,p),base,2) ; for j from 1 to nops(pow2) do if op(j,pow2) <> 0 then a := min(a,op(1,p)^(2^(j-1))) ; end if; end do: end do: return a ; end proc:
seq(A177333(n),n=2..120) ; # R. J. Mathar, Jun 16 2010

Mathematica

b[n_] :=2^(-1+Position[ Reverse@IntegerDigits[n, 2],?(#==1&)])//Flatten; a[n] := Module[{np = PrimePi[n]}, v=Table[0,{np}]; Do[p = Prime[k]; Do[v[[k]] += IntegerExponent[j, p], {j,2,n}],  {k,1,np}]; Min[(Prime/@Range[np])^(b/@v) // Flatten]]; Array[a, 105, 2] (* Amiram Eldar, Sep 17 2019 *)

Extensions

Corrected from a(10) on and extended beyond a(30) by R. J. Mathar, Jun 16 2010