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

%I A177333 #11 Sep 17 2019 16:25:34
%S A177333 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,
%T A177333 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,
%U A177333 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
%N A177333 Smallest factor in the factorization of n! over distinct terms of A050376.
%D A177333 V. S. Shevelev, Multiplicative functions in the Fermi-Dirac arithmetic, Izvestia Vuzov of the North-Caucasus region, Nature sciences 4 (1996), 28-43 [Russian].
%H A177333 Amiram Eldar, <a href="/A177333/b177333.txt">Table of n, a(n) for n = 2..1000</a>
%H A177333 S. Litsyn and V. S. Shevelev, <a href="http://www.emis.de/journals/INTEGERS/papers/h33/h33.Abstract.html">On factorization of integers with restrictions on the exponent</a>, INTEGERS: Electronic Journal of Combinatorial Number Theory, 7 (2007), #A33, 1-36.
%e A177333 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.
%p A177333 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:
%p A177333 seq(A177333(n),n=2..120) ; # _R. J. Mathar_, Jun 16 2010
%t A177333 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 *)
%Y A177333 Cf. A050376, A177329, A055460, A176525, A001358, A176472, A176509, A064380, A050292.
%K A177333 nonn
%O A177333 2,1
%A A177333 _Vladimir Shevelev_, May 06 2010
%E A177333 Corrected from a(10) on and extended beyond a(30) by _R. J. Mathar_, Jun 16 2010