[QUOTE=kriesel;621679]No, I think he's gone fishing plenty already, or at least cast quite widely
<snip>[/QUOTE]My reference was to a card game.

[QUOTE=Dr Sardonicus;621684]My reference was to a card game.[/QUOTE]Of course. And my point was he's already drawn in a sense, an unusually large fraction of the unplayed deck for this game.

[QUOTE=kriesel;621689]Of course. And my point was he's already drawn in a sense, an unusually large fraction of the unplayed deck for this game.[/QUOTE]

You ruin jokes in spectacular ways.

[QUOTE=kriesel;621668]Of the 12 above, 3 now have factors found, 9 have prp/proof & successful cert completed. No fish.[/QUOTE]
Thank you, [COLOR="Blue"]Kriesel[/COLOR], for your [B][COLOR="Red"]awesome[/COLOR][/B] contribution to the computational work on l[COLOR="blue"]ocal wavefronts[/COLOR] of prime exponents!

While still considering the applicability of the standard [COLOR="blue"]Ulam spiral[/COLOR] versus the [COLOR="blue"]Ulam spiral of prime numbers[/COLOR],
meanwhile I intend to concentrate on [COLOR="blue"]TF[/COLOR] and [COLOR="blue"]P-1[/COLOR] tests of a few hundred [COLOR="blue"]28-bit [/COLOR]prime exponents at the [COLOR="Red"]LCS ∩ BSE[/COLOR] intersection (listed below).
[code]
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[/code]
Of them, I further shortlisted 22 prime exponents (listed below) of [COLOR="Blue"]Hamming weight[/COLOR] = [COLOR="Blue"]16[/COLOR], the only [COLOR="blue"]Hamming weight[/COLOR] ≤ 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 [url]https://mersenneforum.org/showpost.php?p=583351&postcount=43[/url].
[code]144542399, 160026343, 162610087, 165179869, 167561501, 182371613, 185958217, 190142417, 190178867, 192514471, 220564159, 226304219, 226308679, 231683267, 242846119, 245205563, 247812911, 247813463, 247814041, 247816879, 256686869, 256796957[/code]

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

Then [I]𝜔[/I]([I]𝜋[/I]([I]p[/I])) 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 [I]p[/I] = 82589933: [I]𝜋[/I]([I]p[/I]) = 4811740 = 2[SUP]2[/SUP] × 5 × 240587, and [I]𝜔[/I]([I]𝜋[/I]([I]p[/I])) = 3.

The histogram of [I]𝜔[/I]([I]𝜋[/I]([I]p[/I])) from 0 to 8 for all 50847534 primes <10[SUP]9[/SUP] is
{1,3050022,11914315,17936312,12876751,4408426,633544,28013,150},

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

The list
{3,4,6,8,9,9,[COLOR="Red"]7[/COLOR],13,11,12,13,[COLOR="red"]8[/COLOR],17,16,18,19,19,[COLOR="red"]12[/COLOR],19,25,23,22,26,26,22,[COLOR="red"]20[/COLOR],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).