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A331284 Number of values of k, 1 <= k <= n, with A329605(k) = A329605(n), where A329605 is the number of divisors of primorial inflation of n (A108951).

%I A331284 #12 Jan 12 2022 03:21:16
%S A331284 1,1,1,1,1,1,1,2,1,1,1,2,1,1,1,1,1,2,1,2,1,1,1,1,1,1,3,2,1,2,1,2,1,1,
%T A331284 1,1,1,1,1,1,1,2,1,2,3,1,1,3,1,2,1,2,1,2,1,1,1,1,1,1,1,1,3,1,1,2,1,2,
%U A331284 1,2,1,2,1,1,3,2,1,2,1,3,1,1,1,1,1,1,1,1,1,2,1,2,1,1,1,1,1,2,3,1,1,2,1,1,3
%N A331284 Number of values of k, 1 <= k <= n, with A329605(k) = A329605(n), where A329605 is the number of divisors of primorial inflation of n (A108951).
%C A331284 Ordinal transform of A329605, or equally, of A329606.
%H A331284 Antti Karttunen, <a href="/A331284/b331284.txt">Table of n, a(n) for n = 1..65537</a>
%H A331284 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%H A331284 <a href="/index/Pri#primorial_numbers">Index entries for sequences related to primorial numbers</a>
%F A331284 a(A331285(n)) = n for all n.
%t A331284 c[n_] := c[n] = If[n == 1, 1, Module[{f = FactorInteger[n], p, e}, If[Length[f] > 1, Times @@ c /@ Power @@@ f, {{p, e}} = f; Times @@ (Prime[Range[PrimePi[p]]]^e)]]];
%t A331284 A329605[n_] := DivisorSigma[0, c[n]];
%t A331284 Module[{b}, b[_] = 0;
%t A331284 a[n_] := With[{t = A329605[n]}, b[t] = b[t] + 1]];
%t A331284 Array[a, 105] (* _Jean-François Alcover_, Jan 12 2022 *)
%o A331284 (PARI)
%o A331284 up_to = 65537;
%o A331284 ordinal_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), pt); for(i=1, length(invec), if(mapisdefined(om,invec[i]), pt = mapget(om, invec[i]), pt = 0); outvec[i] = (1+pt); mapput(om,invec[i],(1+pt))); outvec; };
%o A331284 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));
%o A331284 v331284 = ordinal_transform(vector(up_to, n, A329605(n)));
%o A331284 A331284(n) = v331284[n];
%Y A331284 Cf. A000005, A108951, A329605, A329606, A331285 (positions of the first occurrences of each n, also positions of records).
%Y A331284 Cf. also A067004.
%K A331284 nonn
%O A331284 1,8
%A A331284 _Antti Karttunen_, Jan 14 2020