%I #23 Sep 17 2019 18:50:15
%S 1,2,2,2,2,2,5,5,2,2,7,7,4,4,4,2,2,2,5,5,7,4,3,3,4,4,2,2,4,4,4,4,2,2,
%T 3,3,4,4,3,2,4,4,3,3,2,4,5,5,4,4,2,2,2,2,6,5,2,2,3,3,7,7,3,2,2,2,3,3,
%U 3,4,3,3,4,4,2,2,2,2,6,6,4,4,2,2,2,3,4
%N Number of factorials k!, 0<=k<=n, relatively prime to n! in Fermi-Dirac arithmetic.
%C Or, equivalently, the number of factorials k!, 0<=k<=n, for which k! and n! have no common A050376-factors in their factorizations over distinct terms of A050376.
%C Note that 1 (=0!=1!) corresponds to an empty subset of A050376.
%D V. S. Shevelev, Multiplicative functions in the Fermi-Dirac arithmetic, Izvestia Vuzov of the North-Caucasus region, Nature sciences 4 (1996), 28-43 (Russian; MR 2000f: 11097, pp. 3912-3913).
%H Amiram Eldar, <a href="/A241148/b241148.txt">Table of n, a(n) for n = 0..500</a>
%H 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 0!=1, 1!=1; further we have the following factorizations of k! over distinct terms of A050376 for k = 2,3,4,5,6:
%e 2!=2, 3!=2*3, 4!=2*3*4, 5!=2*3*4*5, 6!=5*9*16. Thus, in the sense of the factorizations being considered, 6! is relatively prime to 0!,1!,2!,3!, and 4!. So a(6)=5.
%t b[n_] := 2^(-1 + Position[Reverse@IntegerDigits[n, 2], _?(# == 1 &)]) // Flatten; infp[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}]; (Prime /@ Range[np])^(b /@ v) // Flatten]; infCoprimeQ[x_, y_] := Intersection[infp[x], infp[y]] == {}; a[n_] := Length @ Select[Range[0, n], infCoprimeQ[n, #] & ]; Array[a, 87, 0] (* _Amiram Eldar_, Sep 17 2019 *)
%Y Cf. A177329, A177333, A177334, A240537, A240606, A240619, A240620, A240668, A240669, A240670, A240672, A240695, A240751, A240755, A240764, A240905, A240906, A241123, A241124, A241139.
%K nonn
%O 0,2
%A _Vladimir Shevelev_, Apr 16 2014
%E More terms from _Peter J. C. Moses_, Apr 18 2014