A341119 - OEIS

%I #29 Nov 04 2023 12:31:06

%S 1,3,6,18,12,36,24,48,72,120,168,336,240,540,360,960,840,1080,720,

%T 1680,3024,1440,2160,2880,2520,6480,4320,14040,8640,5040,9240,7560,

%U 23520,12600,18480,10080,33600,22680,15120,20160,36960,27720,47880,40320,37800,47520,30240,80640,85680,65520,60480

%N a(n) is the least positive number that has exactly n divisors d such that d-1 is prime.

%C a(n) is the least positive solution to A072627(k) = n.

%C The conjectured terms are exact if for 0 <= n <= 10000 we have a(n) / A046523(A000005(a(n))) <= 9. For the found terms, a(n) / A046523(A000005(a(n))) <= 7.3. - _David A. Corneth_, Jun 15 2022

%H Robert G. Wilson v, <a href="/A341119/b341119.txt">Table of n, a(n) for n = 0..1740</a> (first 216 terms from _Robert Israel_)

%H David A. Corneth, <a href="/A341119/a341119.gp.txt">Conjectured first 10001 terms</a>.

%e a(3) = 18 has 3 such divisors: 2+1=3, 5+1=6, 17+1=18, and is the least number with exactly 3.

%p f:= proc(n) nops(select(t -> isprime(t-1), numtheory:-divisors(n))) end proc:

%p N:= 60: count:= 0:

%p V:= Array(0..N):

%p for n from 1 while count < N+1 do

%p   v:= f(n);

%p   if v <= N and V[v] = 0 then

%p     count:=count+1;

%p     V[v]:= n;

%p   fi;

%p od:

%p convert(V,list);

%t With[{s = Array[DivisorSum[#, 1 &, PrimeQ[# - 1] &] &, 10^5]}, Array[FirstPosition[s, #][[1]] &, 51, 0]] (* _Michael De Vlieger_, Feb 05 2021 *)

%o (PARI) a(n) = my(k=1); while (sumdiv(k, d, isprime(d-1)) != n, k++); k; \\ _Michel Marcus_, Feb 05 2021

%Y Cf. A000005, A008864, A046523, A067846, A072627, A341099.

%K nonn

%O 0,2

%A _J. M. Bergot_ and _Robert Israel_, Feb 05 2021