hihi - yes, understood, that was just my equivalent computational *elimination *rule - in fact, if every prime dividing a(n) also divides a(n-1), then there is no valid a(n+1). because every non-trivial common factor between any prospective a(n+1) and a(n) will have at least one of those primes, so the prospective a(n+1) will also have a non-trivial common factor with a(n-1), and is therefore invalid - that means that any prospective value for a(n+1) needs to have at least one prime that does not divide a(n) (or else there is no valid a(n+2)) that made the computation simpler to program more later, chris On Wed, Aug 19, 2020 at 8:08 AM Neil Sloane <njasloane@gmail.com> wrote:

Chris, the definition of Enots Wolley (A336957) is Lexicographically earliest sequence {a(n)} of distinct positive numbers such that, for n>2, a(n) has a common factor with a(n-1) but not with a(n-2). It doesn't say that every prime dividing a(n) divides a(n-1). Look at Theorem 1 in A336957!

Best regards Neil

Neil J. A. Sloane, President, OEIS Foundation. 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com

On Tue, Aug 18, 2020 at 7:10 AM Christopher Landauer <topcycal@gmail.com> wrote:

...

Hihi - If every prime dividing a[n] also divides a[n-1], then there is no integer b that has gcd(b, a[n])>1 and gcd(b, a[n-1])=1, so a[n+1] does not exist

We have a[3]=6, so this criterion applies to b=9

more later, Chris

Sent from my iPhone

...

On Aug 18, 2020, at 02:00, Neil Sloane <njasloane@gmail.com> wrote:

if a(4) = 9 then a(5) does not exist. therefore a(4) > 9

Best regards Neil

Neil J. A. Sloane, President, OEIS Foundation. 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com

On Tue, Aug 18, 2020 at 2:47 AM Frank Stevenson < frankstevensonmobile@gmail.com> wrote:

...

Hi Niel,

I was trying to compute this sequence, but I am having problems understanding the concept of lexicographical earliest. Why is not 9 the 4th number in the sequence, but 15 is ? Why not 105, 1005, 100005 etc ? What am I missing ?

Regards, Frank

...

On Sun, Aug 16, 2020 at 5:56 AM Neil Sloane <njasloane@gmail.com> wrote:

Obviously this is an inverted version of the Yellowstone sequence A098550 ! The name Enots Wolley is for personal use only, it must not be mentioned in the OEIS! We frown on such made-up names.

Definition: Lexicographically earliest sequence {a(n)} of distinct positive numbers such that, for n>2, a(n) has a common factor with a(n-1) but not with a(n-2). 1, 2, 6, 15, 35, 14, 12, 33, 55, 10, 18, 21, 77, 22, 20, 45, 39, 26, 28, 63, ... The original idea was due to Scott, with a different sequence, but this is my (canonical!) version.

Could someone please prove the conjecture that this is a permutation of the set {1, all numbers with at least two distinct prime factors} ?

I can't even prove that every number 2*p (p prime) appears, or that there are infinitely many even terms (although I've found a dozen false proofs). It's a slippery problem.

Neil _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun

_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun

_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun

_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun

-- dr. christopher landauer topcy house consulting thousand oaks, california

I did finish my computation of the 11M first items ( http://traxme.net/11m.zip first 3 are missing from the output ) which should consist of all terms under 100M. I have also looked at the problem reduced to the prime bases [2,3,5,7,11] - which has a much more pronounced holdout effect, where numbers with several smaller factors are picked up late. I have mapped the possible prime base transitions for this problem, and they form a well connected graph with lots of cycles, and crucially the graph has just a single island as opposed to the case for 4 prime bases. This indicates that all combinations of prime bases will be picked up, but with very varying probabilities. This sequence therefore shows a much wider spread: 1, 2, 6, 15, 35, 14, 12, 33, 55, 10, 18, 21, 77, 22, 20, 45, 63, 28, 40, 75, 99, 44, 50, 105, 231, 88, 80, 135, 147, 56, 100, 165, 189, 98, 110, 225, 441, 112, 160, 275, 297, 24, 70, 385, 363, 36, 140, 539, 891, 30, 175, 847, 66, 60, 245, 3773, 132, 90, 875, 5929, 176, 48, 315, 605, 242, 42, 375, 1375, 154, 54, 405, 1225, 196, 72, 495, 1715, 224, 96, 675, 1925, 308, 108, 1125, 2695, 352, 144, 525, 3025, 484, 84, 735, 6655, 704, 126, 945, 6875, 968, 162, 567, 4235, 200, 192, 693, 4375, 250, 198, 1029, 6125, 220, 216, 1323, 8575, 320, 264, 1617, 12005, 400, 288, 1089, 9317, 280, 120, 2673, 26411, 350, 150, 3267, 41503, 392, 180, 825, 65219, 448, 240, 1485, 102487, 686, 270, 1815, 184877, 784, 300, 2475, 290521, 896, 324, 1215, 9625, 616, 384, 1875, 13475, 1078, 432, 2025, 15125, 1232, 168, 3375, 33275, 1408, 252, 1575, 34375, 1936, 294, 2205, 73205, 2662, 336, 2625, 75625, 2816, 378, 2835, 166375, 3872, 486, 1701, 18865, 440, 576, 3087, 21175, 500, 648, 2079, 21875, 640, 396, 2541, 30625, 800, 528, 3969, 42875, 550, 594, 5103, 60025, 880, 726, 7203, 84035, 1000, 768, 3993, 29645, 490, 864, 8019, 46585, 560, 972, 9801, 48125, 700, 1152, 11979, 67375, 980, 1296, 24057, 94325, 1120, 1458, 29403, 105875, 1250, 1536, 4851, 109375, 1280, 792, 6237, 153125, 1600, 1056, 7623, 214375, 2000, 1188, 9261, 300125, 1100, 1452, 11907, 420175, 1210, 1584, 15309, 546875, 1760, 1728, 21609, 132055, 2200, 1944, 27783, 148225, 2420, 2304, 35721, 207515, 2500, 2592, 11319, 171875, 2560, 504, 14553, 366025, 3200, 588, 17787, 378125, 4000, 672, 18711, 805255, 5000, 756, 22869, 831875, 5120, 882, 27951, 859375, 6250, 1008, 33957, 1830125, 6400, 1134, 35937, 1890625, 1400, 1176, 43923, 4026275, 1750, 1344, 72171, 4159375, 1960, 1512, 88209, 4296875, 2240, 1764, 107811, 8857805, 2450, 2016, 131769, 9150625, 2800, 2058, 216513, 9453125, 3430, 2268, 264627, 20131375, 3500, 2352, 323433, 20796875, 3920, 2646, 395307, 21484375, 4480, 2688, 483153, 44289025, 4900, 2916, 649539, 232925, 1372, 3072, 3645, 240625, 1568, 3456, 4125, 326095, 1792, 3888, 4455, 336875, 2744, 4374, 5445, 456533, 3136, 360, 7425, 717409, 3584, 450, 9075, 1127357, 4802, 480, 12375, 1294139, 5488, 540, 13365, 2033647, 6272, 600, 16335, 3195731, 7168, 720, 19965, 5021863, 9604, 750, 20625, 7891499, 10976, 810, 22275, 9058973, 12544, 900, 27225, 12400927, 14336, 960, 37125, 14235529, 19208, 1080, 40095, 22370117, 21952, 1200, 45375, 35153041, 25088, 1350, 49005, 55240493, 28672, 1440, 59895, 63412811, 33614, 1500, 61875, 86806489, 38416, 1620, 66825, 99648703, 43904, 1800, 81675, 136410197, 50176, 1920, 99825, 156590819, 57344, 2160, 103125, 246071287, 67228, 2250, 111375, 386683451, 76832, 2400, 120285, 443889677, 87808, 2430, 136125, 607645423, 100352, 2700, 147015, 697540921, 114688, 2880, 179685, 954871379, 134456, 3000, 185625, 1096135733, 153664, 3240, 200475, 1500512167, 175616, 3600, 219615, 1722499009, 200704, 3750, 226875, 2706784157, 229376, 3840, 245025, 3107227739, 235298, 4050, 299475, 4253517961, 268912, 4320, 309375, 4882786447, 307328, 4500, 334125, 6684099653, 351232, 4608, 5625, 471625, 1694, 5184, 6075, 512435, 2156, 5832, 9375, 529375, 2464, 6144, 10125, 588245, 3388, 1782, 10935, 765625, 4312, 2112, 16875, 1071875, 4928, 2178, 18225, 1500625, 6776, 2376, 28125, 2100875, 7546, 2904, 30375, 2734375, 8624, 3168, 32805, 2941225, 9856, 3564, 46875, 3828125, 11858, 4224, 50625, 4117715, 13552, 4356, 54675, 5359375, 15092, 4752, 84375, 7503125, 17248, 5346, 91125, 10504375, 18634, 5808, 98415, 13671875, 19712, 6336, 140625, 14706125, 23716, 6534, 151875, 19140625, 27104, 6912, 164025, 660275, 5324, 7776, 3675, 741125, 5632, 8748, 4725, 924385, 7744, 9216, 5145, 1037575, 10648, 10368, 6615, 1164625, 11264, 11664, 7875, 1203125, 15488, 12288, 8505, 1452605, 21296, 13122, 11025, 1630475, 22528, 13824, 13125, 1684375, 29282, 15552, 14175, 2282665, 30976, 17496, 15435, 2358125, 42592, 18432, 18375, 2562175, 45056, 20736, 19845, 2646875, 58564, 23328, 23625, 3301375, 61952, 24576, 25515, 3587045, 85184, 26244, 25725, 3705625, 90112, 27648, 33075, 4621925, 117128, 31104, 36015, 5187875, 123904, 34992, 39375, 5636785, 170368, 36864, 42525, 5823125, 180224, 39366, 45927, 6015625, 2750, 41472, 50421, 6470695, 3520, 46656, 64827, 7263025, 4400, 49152, 83349, 8152375, 4840, 52488, 107163, 8421875, 5500, 55296, 137781, 10168235, 6050, 62208, 151263, 11413325, 7040, 69984, 194481, 11790625, 8000, 73728, 43659, 12810875, 10000, 78732, 53361, 13234375, 10240, 82944, 56133, 15978655, 12500, 93312, 68607, 16506875, 12800, 98304, 79233, 17935225, 16000, 104976, 83853, 18528125, 20000, 110592, 101871, 20588575, 20480, 7128, 124509, 26796875, 25000, 7986, 130977, 28824005, 25600, 8448, 160083, 37515625, 31250, 8712, 168399, 52521875, 32000, 9504, 195657, 68359375, 40000, 10692, 205821, 73530625, 40960, 11616, 237699, 95703125, 50000, 12672, 250047, 102942875, 8800, 13068, 321489, 133984375, 9680, 14256, 352947, 144120025, 11000, 15972, 413343, 187578125, 12100, 16038, 453789, 201768035, 13310, 16896, 583443, 262609375, 13750, 17424, 750141, 341796875, 14080, 19008, 964467, 367653125, 17600, 19602, 1058841, 478515625, 19360, 21384, 1240029, 514714375, 22000, 23232, 1361367, 669921875, 24200, 23958, 1750329, 720600125, 26620, 25344, 2250423, 937890625, 27500, 26136, 2470629, 1008840175, 28160, 28512, 2893401, 1313046875, 30250, 31944, 3176523, 1412376245, 35200, 32076, 3720087, 1708984375, 38720, 33792, 4084101, 1838265625, 44000, 34848, 5250987, 2392578125, 48400, 38016, 6751269, 2573571875, 51200, 39204, 251559, 3349609375, 62500, 42768, 305613, 3603000625, 64000, 46464, 307461, 4689453125, 80000, 47916, 373527, 5044200875, 81920, 48114, 392931, 6565234375, 100000, 50688, 480249, 7061881225, 102400, 52272, 505197, 8544921875, 125000, 57024, 554631, 9191328125, 128000, 58806, 586971, 9886633715, 156250, 63888, 617463, ... Up and to this point 3024, 3402, 3528, 4032, 4116, 5636, ... are the smallest candidates that have not yet been included. But I see very little reason to believe that they will be left out, based on how the process appears to have good quality randomness. And A336957 could also be expected to have the same property of including all possible composite numbers, as the number of bases is infinitely larger. Cheers, Frank On Sat, Aug 22, 2020 at 11:38 PM christopher landauer <topcycal@gmail.com> wrote:

hihi, all -

one of the things i noticed is that a very large fraction (i know better than to say almost all in this group 8-)) of the values a[n] are ``late'', where late means a[n] < n

the opposite ones are early, like 6 and 15

but most of the late ones are not very late, usually within a few percent of n (at least up to n=10^6, and the percentage is also decreasing, irregularly as seems to be the case with all properties related to primes) - i wonder of some kind of density argument might get somewhere

it is also interesting to look at the sequence gcd(a[n],a[n-1]) for n>2, which has a subsequence that seems to be all the primes, all in order (but not consecutive; there are lotsa pesky 2's) with a density that appears to be approaching 0.5; that might be a useful step, to show that every prime p occurs in some a[n] with n < 2*p or so

more soon,

chris

_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun