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A241123 Smallest k such that the factorization of k! over distinct terms of A050376 contains exactly n primes.

%I A241123 #20 Sep 17 2019 18:49:56
%S A241123 2,3,5,13,17,21,23,37,33,42,43,56,59,57,75,84,99,101,105,109,123,119,
%T A241123 133,139,157,162,163,182,186,183,207,208,222,219,235,220,255,257,263,
%U A241123 268,267,303,305,307,316,315,340,344,341,343,383,385,387,397,411,425
%N A241123 Smallest k such that the factorization of k! over distinct terms of A050376 contains exactly n primes.
%D A241123 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 A241123 Amiram Eldar, <a href="/A241123/b241123.txt">Table of n, a(n) for n = 1..500</a>
%H A241123 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 A241123 Factorization of 5! over distinct terms of A050376 is 5! = 2*3*4*5. Thus 5 is the smallest k such that such a factorization contains 3 primes: 2,3,5. So a(3)=5.
%t A241123 f[p_, e_] := Mod[e, 2]; b[1] = 0; b[n_] := Plus @@ (f @@@ FactorInteger[n]); m = 56; v = Table[0, {m}]; c = 0; p = 1; n = 2; While[c < m, p *= n; i = b[p]; If[i <= m && v[[i]] == 0, c++; v[[i]] = n]; n++]; v (* _Amiram Eldar_, Sep 17 2019 *)
%o A241123 (PARI) nbp(n) = {f = factor(n); sum (i=1, #f~, f[i,2] % 2);}
%o A241123 a(n) = {k = 1; while(nbp(k!) != n, k++); k;} \\ _Michel Marcus_, Apr 27 2014
%Y A241123 Cf. A240537, A240606, A240619, A240620, A240668, A240669, A240670, A240672, A240695, A240751, A240755, A240764, A240905, A240906.
%K A241123 nonn
%O A241123 1,1
%A A241123 _Vladimir Shevelev_, Apr 16 2014
%E A241123 More terms from _Peter J. C. Moses_, Apr 17 2014