%I #44 Jan 29 2020 02:56:35
%S 1,2,4,3,8,6,16,4,9,12,32,8,64,24,18,5,128,12,256,16,36,48,512,10,27,
%T 96,16,32,1024,24,2048,6,72,192,54,15,4096,384,144,20,8192,48,16384,
%U 64,32,768,32768,12,81,36,288,128,65536,20,108,40,576,1536,131072,30,262144,3072,64,7,216,96,524288,256,1152,72,1048576,18,2097152,6144,48
%N Number of divisors of A108951(n), where A108951 is fully multiplicative with a(prime(i)) = prime(i)# = Product_{i=1..i} A000040(i).
%H Antti Karttunen, <a href="/A329605/b329605.txt">Table of n, a(n) for n = 1..10000</a>
%H <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%H <a href="/index/Pri#primorial_numbers">Index entries for sequences related to primorial numbers</a>
%F a(n) = A000005(A108951(n)).
%F a(n) >= A329382(n) >= A329617(n) >= A329378(n).
%F A020639(a(n)) = A329614(n).
%F From _Antti Karttunen_, Jan 14 2020: (Start)
%F a(A052126(n)) = A329382(n).
%F a(A002110(n)) = A000142(1+n), for all n >= 0.
%F a(n) > A056239(n).
%F a(A329902(n)) = A002183(n).
%F A000265(a(n)) = A331286(n).
%F gcd(n,a(n)) = A331283(n).
%F If n = p(k1)^e(k1) * p(k2)^e(k2) * p(k3)^e(k3) * ... * p(kx)^e(kx), with p(n) = A000040(n) and k1 > k2 > ... > kx, then a(n) = (1+e(k1))^(k1-k2) * (1+e(k1)+e(k2))^(k2-k3) * ... * (1+e(k1)+e(k2)+...+e(kx))^kx.
%F A000035(a(n)) = A000035(A000005(n)) = A010052(n).
%F (End)
%t Block[{a}, a[n_] := a[n] = Module[{f = FactorInteger[n], p, e}, If[Length[f] > 1, Times @@ a /@ Power @@@ f, {{p, e}} = f; Times @@ (Prime[Range[PrimePi[p]]]^e)]]; a[1] = 1; Array[DivisorSigma[0, a@ #] &, 75]] (* _Michael De Vlieger_, Jan 24 2020, after _Jean-François Alcover_ at A108951 *)
%o (PARI)
%o A034386(n) = prod(i=1, primepi(n), prime(i));
%o A108951(n) = { my(f=factor(n)); prod(i=1, #f~, A034386(f[i, 1])^f[i, 2]) }; \\ From A108951
%o A329605(n) = numdiv(A108951(n));
%o (PARI) A329605(n) = if(1==n,1,my(f=factor(n),e=1,m=1); forstep(i=#f~,1,-1, e += f[i,2]; m *= e^(primepi(f[i,1])-if(1==i,0,primepi(f[i-1,1])))); (m)); \\ _Antti Karttunen_, Jan 14 2020
%o (PARI) A329605(n) = if(1==n,1,my(f=factor(n),e=0,d); forstep(i=#f~,1,-1, e += f[i,2]; d = (primepi(f[i,1])-if(1==i,0,primepi(f[i-1,1]))); f[i,1] = (e+1); f[i,2] = d); factorback(f)); \\ _Antti Karttunen_, Jan 14 2020
%Y Cf. A000005, A000040, A000142, A002110, A010052, A034386, A052126, A056239, A108951, A329902, A329378, A329382, A329614, A329617, A331283, A331286 (odd part).
%Y Cf. A329606 (rgs-transform), A329608, A331284 (ordinal transform).
%Y Cf. A331285 (the position where for the first time some term has occurred n times in this sequence).
%K nonn
%O 1,2
%A _Antti Karttunen_, Nov 18 2019