A348184 -id:A348184 - cp's OEIS frontend

A348172 a(n) is the number of positive k (can be greater than n) such that A000005(n)/n = A000005(k)/k.

2, 2, 2, 1, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 1, 2, 3, 2, 1, 2, 2, 2, 3, 2, 2, 2, 1, 2, 2, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 1, 3, 2, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 1, 2, 2, 2, 1, 2, 2, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 1, 2, 2, 2, 2, 2, 2, 3, 1

Offset: 1

Tejo Vrush, Oct 04 2021

The first 1 occurs at a(4). The corresponding k value is 4.
The first 2 occurs at a(1). The corresponding k values are 1, 2.
The first 3 occurs at a(9). The corresponding k values are 9, 18, 24.
The first 4 occurs at a(243). The corresponding k values are 243, 405, 486, 810.
Does every number appear in this sequence?
The first 5 is at a(3189375) with k-values {3189375, 5467500, 6378750, 6561000, 8748000} while the first 6 is at 3176523 with k-values {3176523, 4084101, 6353046, 6806835, 8168202, 13613670}. If 7 appears it does so past 10^8. - Charles R Greathouse IV, Oct 05 2021

			For n = 9, the k such that A000005(9)/9 = 1/3 = A000005(k)/k are 9, 18, and 24. Therefore, a(9) = 3.

David A. Corneth, Table of n, a(n) for n = 1..10000
Charles R Greathouse IV, PARI/GP script

Cf. A000005, A090387, A090395, A005382, A005383, A348184.

Array[Function[r, Count[Range[Ceiling[4/r^2]], _?(DivisorSigma[0, #]/# == r &)]][DivisorSigma[0, #]/#] &, 105] (* or *)
Block[{nn = 7, m, s}, m = 2^(2 nn); s = KeySort@ PositionIndex[Array[DivisorSigma[0, #]/# &, m]]; s = Reverse@ KeyDrop[s, TakeWhile[Keys@ s, 4/#^2 > m &]]; Length /@ Array[Lookup[s, DivisorSigma[0, #]/#] &, 2^nn]] (* Michael De Vlieger, Oct 04 2021 *)

a(n) = {my(q=numdiv(n)/n); sum(i=1, 4/q^2, numdiv(i)/i == q);} \\ Michel Marcus, Oct 04 2021

a(n) = my(q=numdiv(n)/n, s=denominator(q), res = 0); forstep(i=s, 4/q^2, s, if(numdiv(i) == q * i, res++)); res \\ David A. Corneth, Oct 07 2021

PARI
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\\ See Greathouse link
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A349428 Smallest k such that A349410(k) = n or -1 if no such number exists.

Original entry on oeis.org

1, 4, 15, 30, 42, 360, 196, 525, 2080, 320, 7168, 123200, 35200, 150920, 196000, 1232000, 61236, 466560, 106831872, 49787136, 14580000, 155648000, 94058496, 123561984, 47385000

Offset: 1

Tejo Vrush, Nov 17 2021

Cf. A349410.

Similar sequences: A005179, A348184.

f[n_] := Module[{s = NestWhileList[n * DivisorSigma[0, #] &, 1, UnsameQ, All]}, Differences[Position[s, s[[-1]]]][[1, 1]]]; seq[len_, nmax_] := Module[{v = Table[0, {len}], n = 1, c = 0, i}, While[c < len && n < nmax, i = f[n]; If[i <= len && v[[i]] == 0, c++; v[[i]] = n]; n++]; TakeWhile[v, # > 0 &]]; seq[15, 10^6] (* Amiram Eldar, Nov 17 2021 *)

Escape clause value changed to -1. - N. J. A. Sloane, Jan 14 2022

cp's OEIS Frontend

A348172 a(n) is the number of positive k (can be greater than n) such that A000005(n)/n = A000005(k)/k.

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A349428 Smallest k such that A349410(k) = n or -1 if no such number exists.

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