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A187825 Smallest k such that the partial sums of the divisors of k (in decreasing order) generate n primes.

%I A187825 #22 Mar 21 2020 07:12:44
%S A187825 1,3,2,140,560,2160,2772,2016,16830,5148,20592,10640,69300,31200,
%T A187825 156240,177840,288288,143520,927360,1203840,752400,1242360,2702700,
%U A187825 2948400,3996720,1884960,5896800,2692800,1244880,15800400,4586400,11060280,15301440,14414400
%N A187825 Smallest k such that the partial sums of the divisors of k (in decreasing order) generate n primes.
%C A187825 It appears that a(n) is even for n > 0 and nonsquarefree for n > 2. The corresponding triangle of k in which row n gives the n primes starts:
%C A187825 k = 1 -> no prime
%C A187825 k = 3  -> 3;
%C A187825 k = 2  -> 2, 3;
%C A187825 k = 140 -> 293, 307, 317;
%C A187825 k = 560 -> 1373, 1451, 1481, 1487.
%H A187825 Amiram Eldar, <a href="/A187825/b187825.txt">Table of n, a(n) for n = 0..77</a>
%e A187825 a(3) = 140 because the partial sums of the divisors in decreasing order {140, 70, 35, 28, 20, 14, 10, 7, 5, 4, 2, 1} that generate 3 prime numbers are
%e A187825 140 + 70 + 35 + 28 + 20 = 293;
%e A187825 140 + 70 + 35 + 28 + 20 + 14 = 307;
%e A187825 140 + 70 + 35 + 28 + 20 + 14 + 10 = 317.
%p A187825 with(numtheory):for n from 0 to 40
%p A187825 do:ii:=0:for k from 1 to 4000000 while(ii=0) do:s:=0:x:=divisors(k):n1:=nops(x):it:=0:for a from n1 by -1 to 1 do:s:=s+x[a]:if type(s,prime)=true then it:=it+1:else fi:od: if it = n then ii:=1: printf ( "%d %d \n",n,k):else fi:od:od:
%t A187825 a[n_] := Catch[ For[k = 1, True, k++, cnt = Count[ Accumulate[ Divisors[k] // Reverse], _?PrimeQ]; If[cnt == n, Print[{n, k}]; Throw[k]]]]; Table[a[n], {n, 0, 40}] (* _Jean-François Alcover_, Dec 27 2012 *)
%Y A187825 Cf. A023194, A062700, A000203, A187822.
%K A187825 nonn
%O A187825 0,2
%A A187825 _Michel Lagneau_, Dec 27 2012
%E A187825 a(19)-a(33) by _Jean-François Alcover_, Dec 28 2012