A364768 The smallest number k that has exactly n of its divisors in A005153.

1, 2, 4, 8, 12, 32, 24, 60, 48, 72, 96, 120, 144, 420, 384, 240, 432, 360, 576, 480, 864, 840, 1200, 720, 1728, 1800, 4080, 1920, 2400, 1440, 4752, 2160, 3960, 2520, 3600, 2880, 5280, 3360, 9504, 4320, 9240, 5760, 12240, 7200, 7920, 5040, 10800, 8640, 19800, 12600

Offset

1,2

Comments

For any n >= 1, the number 2^(n - 1) has exactly n divisors in A005153.

Links

Table of n, a(n) for n=1..50.

Example

a(1) = 1 because 1 has only one divisor 1 = A005153(1).
a(2) = 2 because 2 has exactly two divisors 1 = A005153(1) and 2 = A005153(2).
Numbers 3, 5, 7, 9, 11 have only divisor 1 in A005153, 4 has divisors 1, 2, 4 in A005153, numbers 6 and 10 have only two divisors in A005153, and 8 has three divisors in A005153. The number 12 has the divisors {1, 2, 3, 4, 6, 12} and exactly five of them, 1, 2, 4 and 12 are in A005153, so a(5) = 12.

Mathematica

f[p_, e_] := (p^(e + 1) - 1)/(p - 1); pracQ[n_] := (ind = Position[(fct = FactorInteger[n])[[;; , 1]]/(1 + FoldList[Times, 1, f @@@ Most@fct]), _?(# > 1 &)]) == {}; d[n_] := DivisorSum[n, 1 &, pracQ[#] &]; seq[len_, nmax_] := Module[{s = Table[0, {len}], c = 0, n = 1, i}, While[c < len && n < nmax, i = d[n]; If[i <= len && s[[i]] == 0, c++; s[[i]] = n]; n++]; s]; seq[50, 10^5] (* Amiram Eldar, Aug 21 2023 *)

Prog

(Magma) sk:=func<n, k|&+[Divisors(n)[i]:i in [1..k]]>; pr:=func<n|forall{k:k in [2..#Divisors(n)]|sk(n, k-1) ge Divisors(n)[k]-1}>; a:=[]; for n in [1..50] do k:=1; while #[d:d in Divisors(k)|pr(d)] ne n  do k:=k+1; end while; Append(~a, k); end for; a;

Crossrefs

Cf. A000005, A005153.

Sequence in context: A358512 A036493 A082906 * A204088 A187941 A085083

Adjacent sequences: A364765 A364766 A364767 * A364769 A364770 A364771

Keyword

nonn

Author

Marius A. Burtea, Aug 18 2023

Status

approved