Exponents of Known Mersenne Primes (Numerical Observations)

[Dr Sardonicus]

Originally Posted by kriesel

No, I think he's gone fishing plenty already, or at least cast quite widely
<snip>

My reference was to a card game.

[kriesel]

Originally Posted by Dr Sardonicus

My reference was to a card game.

Of course. And my point was he's already drawn in a sense, an unusually large fraction of the unplayed deck for this game.

[VBCurtis]

Originally Posted by kriesel

Of course. And my point was he's already drawn in a sense, an unusually large fraction of the unplayed deck for this game.

You ruin jokes in spectacular ways.

[Dobri]

Originally Posted by kriesel

Of the 12 above, 3 now have factors found, 9 have prp/proof & successful cert completed. No fish.

Thank you, Kriesel, for your awesome contribution to the computational work on local wavefronts of prime exponents!

While still considering the applicability of the standard Ulam spiral versus the Ulam spiral of prime numbers,
meanwhile I intend to concentrate on TF and P-1 tests of a few hundred 28-bit prime exponents at the LCS ∩ BSE intersection (listed below).

134319307, 135113387, 135226027, 137421587, 137443751, 137465033, 137767621, 137767681, 141657757, 141662617, 141677359, 144335641, 144491777, 144542399, 144759757, 148552753, 148561201, 149489201, 149489573, 149523601, 149526683, 149539889, 149542973, 149546641, 149865241, 154544177, 154544837, 158739221, 159414859, 160026343, 160039699, 160053857, 160190183, 160915879, 160916083, 160985497, 160996813, 161057191, 161543399, 161880113, 161899289, 162587441, 162610087, 162611623, 165179869, 167561501, 167571071, 167880563, 168024221, 168215893, 168659627, 174076219, 174113561, 174238769, 174266777, 174371143, 175515661, 176625821, 177736753, 177737551, 177760733, 177761621, 177762769, 177773333, 177782833, 178882097, 178955779, 178955879, 178962481, 179337787, 180587569, 182371613, 184365449, 184367893, 184445213, 185356007, 185436841, 185861789, 185864051, 185947441, 185948209, 185948317, 185958217, 185958413, 185958593, 190142417, 190178867, 190195391, 190195519, 192289891, 192292549, 192292669, 192514471, 192615811, 193341299, 193344257, 193906003, 194706019, 195342533, 196137937, 197335433, 197938883, 198174953, 204522781, 204549197, 204571417, 206385379, 206408387, 206488753, 214034921, 214216901, 214328089, 215193757, 215391871, 220445461, 220564159, 221250329, 221904157, 221975893, 222015809, 222101137, 223104619, 223750193, 226304219, 226308119, 226308419, 226308679, 226523747, 226695577, 227134513, 227135347, 227148199, 227150851, 227151139, 227162723, 227162899, 227191571, 231493351, 231493357, 231654751, 231683267, 232288643, 232288649, 232299913, 236466403, 236609959, 236610329, 236656009, 238752781, 242704433, 242846119, 244349543, 245205563, 245205673, 247479601, 247771333, 247780249, 247786571, 247786573, 247792009, 247793347, 247799077, 247799177, 247812911, 247813463, 247814041, 247814243, 247816147, 247816879, 255099673, 255530377, 256392497, 256436611, 256436617, 256686869, 256796957, 256829119, 256849319, 259401841, 261659843, 261776153, 265048291, 265534229

Of them, I further shortlisted 22 prime exponents (listed below) of Hamming weight = 16, the only Hamming weight ≤ 17 of a prime exponent for which no Mersenne prime has been found (see the attached histogram).
Note: An initial version of the attached histogram can be found at https://mersenneforum.org/showpost.p...1&postcount=43.

144542399, 160026343, 162610087, 165179869, 167561501, 182371613, 185958217, 190142417, 190178867, 192514471, 220564159, 226304219, 226308679, 231683267, 242846119, 245205563, 247812911, 247813463, 247814041, 247816879, 256686869, 256796957

[Dobri]

Let 𝜋(p) be the the prime-counting function,
and 𝜔(n) be the prime (little) omega function
which counts the number of distinct prime factors in the integer n.

Then 𝜔(𝜋(p)) of the 51 exponents of known Mersenne primes is given as
{0,1,1,1,2,1,1,1,2,2,2,1,2,2,2,2,2,3,2,3,3,2,2,3,2,1,2,2,2,3,4,2,2,3,2,4,2,3,2,3,3,3,4,3,3,3,4,4,3,4,3}.

For example, for p = 82589933: 𝜋(p) = 4811740 = 2² × 5 × 240587, and 𝜔(𝜋(p)) = 3.

The histogram of 𝜔(𝜋(p)) from 0 to 8 for all 50847534 primes <10⁹ is
{1,3050022,11914315,17936312,12876751,4408426,633544,28013,150},

and the histogram of 𝜔(𝜋(p)) from 0 to 8 for the 51 exponents of known Mersenne primes is
{1,8,21,15,6,0,0,0,0}.

[Dobri]

The list
{3,4,6,8,9,9,7,13,11,12,13,8,17,16,18,19,19,12,19,25,23,22,26,26,22,20,50}
gives the sum of 1s at distinct bit positions from 1 to 27 for the 27-bit (with padding of 0s on the left) binary representation of the 51 known prime exponents of known Mersenne primes (see also the attached bar graph).
The leftmost bit position 1 corresponds to the most significant bit (MSB) and the rightmost bit position 27 corresponds to the least significant bit (LSB).