cp's OEIS Frontend

A069784 Numbers m such that gcd(d((m!)^3), d(m!)) = 2^k, i.e., is a power of 2; d = A000005.

%I A069784 #34 Aug 04 2017 15:39:23
%S A069784 1,2,3,4,5,6,7,8,9,12,13,14,18,19,30,31,32,35,38,39
%N A069784 Numbers m such that gcd(d((m!)^3), d(m!)) = 2^k, i.e., is a power of 2; d = A000005.
%C A069784 From _David A. Corneth_, Jul 31 2017: (Start)
%C A069784 Theorem: There are no further terms.
%C A069784 Proof:
%C A069784 Let e_n(n, p) be the exponent of p in n!. The prime p has exponent e_(n, p) = n/p for sqrt(n) < p < n in n!. n/4 <= p < n/3, e_(n, p) = 3 so e_(n, p) * 3 = 9. and for n/5 <= p < n/4, e = 4. The gap g_n between prime(n) and prime(n+1) is about sqrt(n) * log(n). There is a gap of n/4 - n/5 = n/20 between n/5 and n/4. primepi(1000) = 168, so for n > 5*1000, the gap between n/5 and the next prime is about sqrt(168) * log(168) ~= 66. This is much less than n/20. No 40 <= m <= 15000 is in the sequence, which completes the proof. (End)
%t A069784 Do[s=GCD[DivisorSigma[0, (n!)^3], DivisorSigma[0, n! ]]; If[IntegerQ[n/100], Print[{n}]]; If[IntegerQ[Log[2, s]], Print[n]], {n, 1, 10000}]
%o A069784 (PARI) val(n, p) = my(r=0); while(n, r+=n\=p);r
%o A069784 is(n) = {my(p1 = p2 = 1); forprime(p=2, n, v = val(n, p); p1 *= (v + 1); p2 *= (3*v + 1)); g = gcd(p1, p2); g==2^(valuation(g, 2))} \\ _David A. Corneth_, Jul 31 2017
%Y A069784 Cf. A000005, A000142, A069780, A069781, A069782, A069783.
%K A069784 nonn,fini,full
%O A069784 1,2
%A A069784 _Labos Elemer_, Apr 08 2002
%E A069784 Keywords fini and full added by _David A. Corneth_, Jul 31 2017