A331284 - OEIS

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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).

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

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OFFSET

1,8

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

FORMULA

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

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 *)

PROG

(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];

CROSSREFS

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

Cf. also A067004.

Sequence in context: A210960 A339095 A193509 * A331591 A003649 A353741

Adjacent sequences: A331281 A331282 A331283 * A331285 A331286 A331287

KEYWORD

nonn

AUTHOR

Antti Karttunen, Jan 14 2020

STATUS

approved