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A357332 2-adic valuation of A000793(n).

%I A357332 #12 Sep 25 2022 04:25:25
%S A357332 0,1,0,2,1,1,2,0,2,1,1,2,2,2,0,2,1,1,2,2,2,2,3,3,2,2,2,1,3,2,2,2,2,3,
%T A357332 3,2,2,2,2,3,1,3,2,2,2,2,3,3,2,2,2,2,3,3,3,3,3,1,3,2,2,2,2,3,3,2,2,2,
%U A357332 2,3,3,3,3,3,3,3,1,4,2,2,2,2,3,3,2,2,2,2,3,3,3
%N A357332 2-adic valuation of A000793(n).
%C A357332 Is it true that lim_{n->+oo} a(n) = +oo? It seems that the last occurrences of 0, 1, 2, 3, and 4 appear at indices 15, 77, 667, 4535, and 7520. More generally, is it true that lim_{n->+oo} v(A000793(n),p) = +oo for every prime p, where v(k,p) is the p-adic valuation of k?
%H A357332 Jianing Song, <a href="/A357332/b357332.txt">Table of n, a(n) for n = 1..10000</a>
%e A357332 a(15) = 0 since A000793(15) = lcm(3,5,7) = 105 is odd.
%e A357332 a(77) = 1 since A000793(77) = lcm(2,3,5,7,11,13,17,19) = 9699690 is even but not divisible by 4.
%o A357332 (PARI) listn(N) = {
%o A357332   my(V = vector(N, n, 1));
%o A357332    forprime (i=2, N,  \\ primes i
%o A357332       forstep (j=N, i,  -1,
%o A357332          my( hi = V[j] );
%o A357332          my( pp = i );  \\ powers of prime i
%o A357332          while ( pp<=j,  \\ V[] is 1-based
%o A357332              hi = max(if(j==pp, pp, V[j-pp]*pp), hi);
%o A357332              pp *= i;
%o A357332          );
%o A357332          V[j] = hi;
%o A357332       );
%o A357332    );
%o A357332    vector(N, n, valuation(V[n], 2));
%o A357332 } \\ copied from _Joerg Arndt_'s code for A000793
%Y A357332 Cf. A000793.
%K A357332 nonn
%O A357332 1,4
%A A357332 _Jianing Song_, Sep 24 2022