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A329614 Smallest prime factor of the number of divisors of A108951(n).

%I A329614 #16 Nov 18 2019 22:04:07
%S A329614 1,2,2,3,2,2,2,2,3,2,2,2,2,2,2,5,2,2,2,2,2,2,2,2,3,2,2,2,2,2,2,2,2,2,
%T A329614 2,3,2,2,2,2,2,2,2,2,2,2,2,2,3,2,2,2,2,2,2,2,2,2,2,2,2,2,2,7,2,2,2,2,
%U A329614 2,2,2,2,2,2,2,2,2,2,2,2,5,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,2,2,2,2,2
%N A329614 Smallest prime factor of the number of divisors of A108951(n).
%C A329614 Differs from A071187 for the first time at n=324, where a(324) = 5, while A071187(324) = 3. The positions of the differences are listed at A329613.
%H A329614 Antti Karttunen, <a href="/A329614/b329614.txt">Table of n, a(n) for n = 1..100000</a>
%H A329614 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%H A329614 <a href="/index/Pri#primorial_numbers">Index entries for sequences related to primorial numbers</a>
%F A329614 a(n) = A071187(A108951(n)).
%F A329614 a(n) = A020639(A329605(n)).
%e A329614 324 = 18^2 = 2^2 * 3^4, thus A108951(324) = 2^2 * (2*3)^4 = 2^6 * 3^4 = 5184, which has (6+1)*(4+1) = 7 * 5 = 35 divisors, thus a(324) = A020639(35) = 5.
%t A329614 Array[FactorInteger[DivisorSigma[0, #]][[1, 1]] &@ Apply[Times, Map[#1^#2 & @@ # &, FactorInteger[#] /. {p_, e_} /; e > 0 :> {Times @@ Prime@ Range@ PrimePi@ p, e}]] &, 105] (* _Michael De Vlieger_, Nov 18 2019 *)
%o A329614 (PARI)
%o A329614 A034386(n) = prod(i=1, primepi(n), prime(i));
%o A329614 A108951(n) = { my(f=factor(n)); prod(i=1, #f~, A034386(f[i, 1])^f[i, 2]) };  \\ From A108951
%o A329614 A071187(n) = if(1==n, n, my(f = factor(numdiv(n))); vecmin(f[, 1]));
%o A329614 A329614(n) = A071187(A108951(n));
%Y A329614 Cf. A020639, A034386, A071187, A108951, A329605, A329608, A329612, A329613.
%K A329614 nonn
%O A329614 1,2
%A A329614 _Antti Karttunen_, Nov 17 2019