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A177334 Largest factor in the factorization of n! over distinct terms of A050376.

%I A177334 #18 Sep 17 2019 16:28:52
%S A177334 2,3,4,5,16,16,16,81,256,256,256,256,256,256,256,256,65536,65536,
%T A177334 65536,65536,65536,65536,65536,65536,65536,65536,65536,65536,65536,
%U A177334 65536,65536,65536,4294967296,4294967296,4294967296,4294967296,4294967296,4294967296
%N A177334 Largest factor in the factorization of n! over distinct terms of A050376.
%C A177334 Each number >=2 has a unique factorization over distinct terms of A050376.
%C A177334 This is obtained from the standard prime factor representation by splitting the exponents into a sum of powers of 2, and further factorization according to the nonzero term of this base-2 representation.
%C A177334 The largest factor of this representation of A000142(n) defines this sequence.
%D A177334 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 A177334 Amiram Eldar, <a href="/A177334/b177334.txt">Table of n, a(n) for n = 2..1000</a>
%H A177334 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.
%p A177334 A177334 := proc(n) local a,p,pow2 ; a := 1 ; 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 := max(a,op(1,p)^(2^(j-1))) ; end if; end do: end do: return a ; end proc:
%p A177334 seq(A177334(n),n=2..60) ; # _R. J. Mathar_, Jun 16 2010
%t A177334 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}]; Max[(Prime/@Range[np])^(b/@v) // Flatten]]; Array[a, 38, 2]  (* _Amiram Eldar_, Sep 17 2019 *)
%Y A177334 Cf. A050376, A177329, A055460, A177333, A176525, A001358, A176509, A064380, A050292.
%K A177334 nonn
%O A177334 2,1
%A A177334 _Vladimir Shevelev_, May 06 2010
%E A177334 a(18) and a(19) corrected and sequence extended by _R. J. Mathar_, Jun 16 2010