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).

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, 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, 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

Offset: 1

Author

Antti Karttunen, Jan 14 2020

Keywords

• nonn

Comments

Ordinal transform of A329605, or equally, of A329606.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537
• Index entries for sequences computed from indices in prime factorization
• Index entries for sequences related to primorial numbers

Crossrefs

Cf. A000005, A108951, A329605, A329606, A331285 (positions of the first occurrences of each n, also positions of records).

Cf. also A067004.

Programs

Mathematica

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)]]];
A329605[n_] := DivisorSigma[0, c[n]];
Module[{b}, b[_] = 0;
a[n_] := With[{t = A329605[n]}, b[t] = b[t] + 1]];
Array[a, 105] (* Jean-François Alcover, Jan 12 2022 *)

PARI

up_to = 65537;
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; };
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));
v331284 = ordinal_transform(vector(up_to, n, A329605(n)));
A331284(n) = v331284[n];

Formula

a(A331285(n)) = n for all n.