Puzzle 1003. Right Perfect Primes

This is an update for an issue that started in our puzzle 203 and continued through the puzzles 858 & 859. This is an issue that cover already 20 years, circa.

The integers there discussed are the so-called "Right Perfect Integers" and the aim is to discover what of these integers are primes.

Let's make a very short & crowded summary of the issue.

No Odd Perfect Integers are known.
Every Even Perfect Integer known is related to one Mersenne prime. If 2^p-1 is a Mersenne prime, then p is prime too and
2^(p-1)*(2^p-1) is an Even Perfect Integer.
What if we add or subtract one unit to an Even Perfect Integer? Is it possible for these new odd integers to be primes?
If we subtract one unit to an Even Perfect Integer, the only possible prime produced is 5, when p=2, 2^(2-1)*(2^2-1)-1=5. For any other p value a composite is produced. See the whole proof in Puzzle 203.

But if we add one unit, 2^(p-1)*(2^p-1)+1 the corresponding Right Perfect Integers (RPI) has shown a more interesting and diverse behavior. Here is a condensed summary of this situation:

Quantity of RPI	Behavior of the RPI	Nth Mersenne prime
1	Primality or Compositeness unknown	49
4	Primes (by E. Labos)	1, 2, 5 &7
2	Composite without known smallest prime factor	15, & 30
10	Composite with known smallest prime factor computed	9, 16, 17, 20, 26, 29, 31, 40, 41 & 46
34	Composite with known smallest prime factor obtained by the Wilke's rules	All the others
51	Totals

*See below some details for all the 51 Mersenne Primes...

The status shown above is product of a collective work of several puzzlers that has sent their contributions to the Puzzles 203, 858 & 859, from 2001 to our days. Here is a short historical summary:

E. Labos, is the creator of the concept RPI. The participants in these three past puzzles are: J. K. Andersen, Jan van Delden, Carlos Rivera, J. C. Rosa, Shyam Sunder Gupta, Emmanuel Vantieghem, Polly T. Wang, Johann Wiesenbauer and Ken Wilke (alphabetic order)

The Wilke's rules are:

if p is a prime of the form 6k+5, then RPI(p) is divisible by 7.
if p is a prime of the form 10k+7, then RPI(p) is divisible by 11
if p is a prime of the form 28k+3, then RPI(p) is divisible by 29
if p is a prime of the form 70k+5, then RPI(p) is divisible by 71.

Until new & interesting (for our purposes) Mersenne primes are gotten, nowadays (May 30, 2020) we have only two pending tasks (in red in the summary above):

a) To determine the primality or compositeness of the RPI associated to the Mersenne prime 49.

b) To get the smallest prime factor (SPF) of the RPI's associated to the Mersenne primes 15th & 30th

Four years ago (2016) Emmanuel Vantieghem and Jan van Delden worked a bit over these two tasks.

Q. Would you like to give a new try to any of these two tasks?

***

Details

Mersenne Primes (Mp), Perfect numbers (Pp) and Right Perfect Integers (Pp+1)

Mp=	2^p-1		Pp=	2^(p-1)*(2^p-1)

	Extract from:	https://primes.utm.edu/mersenne/index.html
#	p	Digits in Mp	Digits in Pp	Status of Pp+1
1	2	1	1	Prime by E. Labos
2	3	1	2	Prime by E. Labos
3	5	2	3	Composite w/SPF by KW rule
4	7	3	4	Composite w/SPF by KW rule
5	13	4	8	Prime by E. Labos
6	17	6	10	Composite w/SPF by KW rule
7	19	6	12	Prime by E. Labos
8	31	10	19	Composite w/SPF by KW rule
9	61	19	37	Composite w/SPF computed
10	89	27	54	Composite w/SPF by KW rule
11	107	33	65	Composite w/SPF by KW rule
12	127	39	77	Composite w/SPF by KW rule
13	521	157	314	Composite w/SPF by KW rule
14	607	183	366	Composite w/SPF by KW rule
15	1279	386	770	Composite w/o SPF known
16	2203	664	1327	Composite w/SPF computed
17	2281	687	1373	Composite w/SPF computed
18	3217	969	1937	Composite w/SPF by KW rule
19	4253	1281	2561	Composite w/SPF by KW rule
20	4423	1332	2663	Composite w/SPF computed
21	9689	2917	5834	Composite w/SPF by KW rule
22	9941	2993	5985	Composite w/SPF by KW rule
23	11213	3376	6751	Composite w/SPF by KW rule
24	19937	6002	12003	Composite w/SPF by KW rule
25	21701	6533	13066	Composite w/SPF by KW rule
26	23209	6987	13973	Composite w/SPF computed
27	44497	13395	26790	Composite w/SPF by KW rule
28	86243	25962	51924	Composite w/SPF by KW rule
29	110503	33265	66530	Composite w/SPF computed
30	132049	39751	79502	Composite w/o SPF known
31	216091	65050	130100	Composite w/SPF computed
32	756839	227832	455663	Composite w/SPF by KW rule
33	859433	258716	517430	Composite w/SPF by KW rule
34	1257787	378632	757263	Composite w/SPF by KW rule
35	1398269	420921	841842	Composite w/SPF by KW rule
36	2976221	895932	1791864	Composite w/SPF by KW rule
37	3021377	909526	1819050	Composite w/SPF by KW rule
38	6972593	2098960	4197919	Composite w/SPF by KW rule
39	13466917	4053946	8107892	Composite w/SPF by KW rule
40	20996011	6320430	12640858	Composite w/SPF computed
41	24036583	7235733	14471465	Composite w/SPF computed
42	25964951	7816230	15632458	Composite w/SPF by KW rule
43	30402457	9152052	18304103	Composite w/SPF by KW rule
44	32582657	9808358	19616714	Composite w/SPF by KW rule
45	37156667	11185272	22370543	Composite w/SPF by KW rule
46	42643801	12837064	25674127	Composite w/SPF computed
47	43112609	12978189	25956377	Composite w/SPF by KW rule
48?	57885161	17425170	34850339	Composite w/SPF by KW rule
49?	74207281	22338618	44677235	Unknow if prime or composite
50?	77232917	23249425	46498850	Composite w/SPF by KW rule
51?	82589933	24862048	49724095	Composite w/SPF by KW rule