hihi, all - here is a table of the smallest acceptable (eligible) value left out when i let the program search up to nmax; each of them is found soon after this is the smallest acceptable value that does not occur for n up to nmax, not the smallest n before which an unacceptable value does not occur (the nmax=1500 line was really confusing until i figured this out; it means that the next value that does occur is at the given n; so for nmax=1500, other acceptable values did not occur, but all were larger than 1458) nmax first missing value where it occurs 100 104 at n=49 n=103 200 204 at n=124 n=203 300 304 at n=224 n=303 400 372 at n=387 n=403 500 490 at n=458 n=503 this is weird 600 588 at n=337 n=607 700 684 at n=481 n=703 800 780 at n=516 n=807 900 858 at n=629 n=907 1000 942 at n=457 n=1003 1100 1068 at n=1078 n=1103 this is really weird 1200 1170 at n=1062 n=1210 1300 1266 at n=372 n=1303 1400 1356 at n=984 n=1411 1500 1458 at n=1499 n=1518 1600 1572 at n=1121 n=1603 1700 1678 at n=989 n=1703 ok, so i suspect my program is broken 1800 1770 at n=1229 n=1810 1900 1866 at n=1830 n=1903 2000 1938 at n=1460 n=2003 3000 2940 at n=2133 n=3007 4000 3894 at n=2970 n=4004 5000 4884 at n=3882 n=5013 6000 5864 at n=4642 n=6001 7000 6876 at n=5543 n=7009 8000 7848 at n=6130 n=8022 9000 8828 at n=7170 n=9002 10000 9790 at n=9993 n=?? curiously, they are all even but i checked the small results and they look right ok, i sort of have an explanation of these weird values - define the ``lateness'' of a[n] as a[n] - n; we know that many times that is positive, but it can also be negative; the a[n] for which a[n] < n are ``late'' (we know that many values occur early, such as 15 and 35); it is a curious phenomena that very few a[n] are very late, and most are not very late at all; it looks like only a few percent asymptotically (n-a[n]) / a[n] for n with n > a[n]; that means that you should expect the ones that were missed up to nmax to be found soon after that; the commonality of the numbers is what is scary the time complexity of the program is roughly O(nmax ^ 2) - not derived, just observed; i also have an observed relationship between nmax and the maximum prospective a[n], which i call bmax; roughly (for large enough n) bmax ~ nmax ^ 1.2 (with the exponent seeming to decrease over time, but slowly) i will have output for nmax=10^6 in a few days various graphs are really helpful more soon, chris On 2020-08-16 15:02, Allan Wechsler wrote:
Chris, what is the smallest eligible (non-prime-power) number your list leaves out?
On Sun, Aug 16, 2020 at 5:55 PM Christopher Landauer <topcycal@gmail.com> wrote:
hihi, all - i wrote a program to generate terms to n=1000 (i still need to verify it, but it matches what was in this e-mail)
it is a curious fact that the gcd of successive terms is a prime, up to a point, and mostly in any case
i'll have more in a few days
more later, chris
On Sun, Aug 16, 2020 at 6:04 AM Fred Lunnon <fred.lunnon@gmail.com> wrote:
Already posted on August 11th ?!
WFL
On 8/16/20, 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.
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-- dr. christopher landauer topcy house consulting thousand oaks, california _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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