20200223, 16:57  #1 
Dec 2008
you know...around...
2×311 Posts 
Gaps in kalmost primes relative to 2^k, a question
Consider gaps in kalmost primes (i.e. (n)=k), relative to the size of the smallest such number, 2^{k}.
Let k, we have the sequence Code:
1 1.5 2.25 2.5 3.375 3.5 3.75 5.0625 5.25 5.5 5.625 6.25 6.5 7.59375 7.875 8.25 8.4375 8.5 8.75 9.375 9.5 9.75 11.390625 11.5 11.8125 12.25 12.375 12.65625 12.75 13.125 13.75 14.0625 14.25 14.5 14.625 15.5 15.625 16.25 17.0859375 17.25 17.71875 18.375 18.5 18.5625 18.984375 19.125 19.25 19.6875 20.5 20.625 21.09375 21.25 21.375 21.5 21.75 21.875 21.9375 22.75 23.25 23.4375 23.5 23.75 24.375 25.62890625 ... Looking for record gaps, we have successive maxima (these, when multiplied by 2^{k}, are eventually the record gaps between kalmost primes for almost all k): Code:
gap after n/2^k= 0.5 1 0.75 1.5 0.875 2.5 1.3125 3.75 1.640625 9.75 1.880859375 36.5625 1.904296875 3506216.064453125 Question is: are those gaps bounded? 
20200329, 20:48  #2 
Dec 2008
you know...around...
1156_{8} Posts 
Update: A gap greater than 2 is possible!
Code:
gap after n/2^k= 0.5 1 0.75 1.5 0.875 2.5 1.3125 3.75 1.640625 9.75 1.880859375 36.5625 1.904296875 3506216.064453125 1.921875 17946832.921875 1.96875 44901649.8984375 2.21875 54738288.6875 Code:
1.9609375 105154231.0078125 Code:
gap after n/2^k= 1/2 1 1/4 2.25 1/8 3.375 1/16 8.4375 1/32 29.5 1/64 82.25 5/2048 86.49755859375 1/512 397.248046875 1/2048 2969.74951171875 1/4096 7943.359130859375 1/8192 39853.75 43/524288 155214.1249179840087890625 1/16384 199841.796875 1/32768 211470.312469482421875 1/131072 287586.8125 1/262144 1799287.749996185302734375 1/524288 22043660.546875 1/2097152 24139755.968749523162841796875 1/4194304 65777924.75 1/33554432 67956505.2499999701976776123046875 A126279 provides enough data to add a few more terms to the former sequence after we calculate how many odd numbers with 50n+k factors there are up to 2^(50+k), for k=1...n*log(3)/log(1.5)50. I think I could do it, but not before mid April, so if anyone is interested... 
20200401, 22:50  #3 
May 2018
2×101 Posts 
Interesting. I wonder why the gap of 1.880859375 from 36.5625 to 38.443359375 stays as a record for so long.

20200402, 19:36  #4  
Dec 2008
you know...around...
1156_{8} Posts 
Quote:
But it's quite a large gap in those gaps indeed. BTW, there was another gap > 2 shortly after my last post. These are the 11 largest gaps for x < 2^27: Code:
decimal or between and 2.21875 71b5 875812619b4 1751625309b5 2.0078125 257b7 973065703b3 15569051505b7 1.96875 63b5 5747411187b7 5747411439b7 1.9609375 251b7 13459741569b7 3364935455b5 1.921875 123b6 1148597307b6 574298715b5 1.904296875 975b9 1569390625b9 98086975b5 1.880859375 963b9 585b4 19683b9 1.875 15b3 459355401b4 459355431b4 1.875 15b3 328429733b2 656859481b3 1.863212... 244215b17 1852276489785b17 115767295875b13 1.84765625 473b8 180278711b3 5768919225b8 If anyone wants to continue the search for [2^27, 2^28], I've attached my Pari program. On my mediocre laptop, it took a little less than six days to search [2^26, 2^27], I expect a fast PC to need about the same time for the larger interval. The program searches in intervals of 2^20, each of them takes about 3+ hours (maybe 1+ hour on a fast PC; the time increases as the numbers get larger), so you'll have to be patient until you see the first output... Last fiddled with by mart_r on 20200402 at 19:52 

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