A187822 -id:A187822 - cp's OEIS frontend

A193934 Triangle read by rows: row n gives the n primes corresponding to A187822.

3, 3, 7, 3, 7, 31, 3, 7, 31, 127, 3, 7, 19, 29, 43, 3, 7, 41, 61, 83, 167, 3, 7, 19, 29, 43, 151, 271, 3, 11, 17, 53, 163, 409, 1109, 1439, 3, 61, 79, 103, 283, 1171, 1459, 3187, 4339, 3, 7, 19, 29, 43, 163, 233, 307, 1039, 1409, 3, 29, 59, 71, 233, 269, 353

Offset: 1

Michel Lagneau, Jan 02 2013

			Triangle begins:
n = 1 and k = 2  ->    [3]
n = 2 and k = 4  ->    [3, 7]
n = 3 and k = 16 ->   [3, 7, 31]
n = 4 and k = 64 ->   [3, 7, 31, 127]
n = 5 and k = 140 -> [3, 7, 19, 29, 43]
n = 6 and k = 440 -> [3, 7, 41, 61, 83, 167]
…
The sequence A187822 gives the values k.

Cf. A187822.

with(numtheory):for n from 0 to 30
do:ii:=0:for k from 1 to 4000000 while(ii=0) do:s:=0:x:=divisors(k):n1:=nops(x):it:=0:lst:={}:for a from 1 to n1 do:s:=s+x[a]:if type(s,prime)=true then it:=it+1:lst:=lst union {s}:else fi:od: if it = n then ii:=1: print(lst) :else fi:od:od:

lst={2};Do[ lst=Union[lst ,{Prime[i]}],{i,1,5000}];a[n_]:=Catch[For[k=1,True,k++,cnt=Count[Accumulate[Divisors[k]],_?PrimeQ];If[cnt==n,Print[Intersection[Accumulate[Divisors[k]],lst]];Throw[k]]]];Table[a[n],{n,0,15}]

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

Original entry on oeis.org

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

Michel Lagneau, Dec 27 2012

nonn

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.

			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.

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

Cf. A023194, A062700, A000203, A187822.

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:

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 *)

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

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A193934 Triangle read by rows: row n gives the n primes corresponding to A187822.

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