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

1, 3, 2, 140, 560, 2160, 2772, 2016, 16830, 5148, 20592, 10640, 69300, 31200, 156240, 177840, 288288, 143520, 927360, 1203840, 752400, 1242360, 2702700, 2948400, 3996720, 1884960, 5896800, 2692800, 1244880, 15800400, 4586400, 11060280, 15301440, 14414400

Offset: 0

Author

Michel Lagneau, Dec 27 2012

Keywords

• nonn

Comments

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:

k = 1 -> no prime

k = 3 -> 3;

k = 2 -> 2, 3;

k = 140 -> 293, 307, 317;

k = 560 -> 1373, 1451, 1481, 1487.

Examples

			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
140 + 70 + 35 + 28 + 20 = 293;
140 + 70 + 35 + 28 + 20 + 14 = 307;
140 + 70 + 35 + 28 + 20 + 14 + 10 = 317.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 0..77

Crossrefs

Cf. A023194, A062700, A000203, A187822.

Programs

Maple

with(numtheory):for n from 0 to 40
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:

Mathematica

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 *)

Extensions

a(19)-a(33) by Jean-François Alcover, Dec 28 2012