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A341652
Numbers j > 1 at which the absolute difference between log(j)*log(A341651(j)) and the nearest integer reaches a record low.
4
2, 3, 4, 6, 11, 12, 53, 60, 146, 315, 328, 391, 438, 509, 659, 883, 1043, 1398, 12381, 41956, 50372, 61220, 81580, 189608, 205666, 224305, 4625460, 4969792, 8505660, 12231434, 20044438, 27091898, 807041817, 2388370134, 4024950319, 5642521468, 5713420381, 9015369803
OFFSET
1,1
COMMENTS
Let t(j,k) = log(j)*log(k) and let d(j,k) be the absolute difference of t(j,k) and the nearest integer, i.e., d(j,k) = |t(j,k) - round(t(j,k))|. This sequence lists the numbers j at which a record low value of d(j,k) (i.e., lower than for any other pair (j',k') satisfying 2 <= k' <= j' <= j) occurs. Corresponding values of k are in A341653.
Viewed on a log-log scatterplot (see links), the records at a(6)=12 (see A341651), a(18)=1398 (see A341577), a(26)=224305, and a(32)=27091898 are among the most noteworthy; at each of these, the value of d(j,k) is smaller than that at the previous record by at least an order of magnitude.
LINKS
Jon E. Schoenfield, Magma program
Jon E. Schoenfield, scatterplot
EXAMPLE
n a(n) A341653(n) d(a(n), A341653(n))
-- ---------- ---------- --------------------------
1 2 2 0.480453013918201424667...
2 3 3 0.206948960812581977843...
3 4 2 0.039093972163597150665...
4 6 3 0.031551028768609035727...
5 11 8 0.013716957145365614933...
6 12 5 0.000697028971586630527...
7 53 34 0.000680675152963635202...
8 60 39 0.000116295803634252268...
9 146 101 0.000054832287899364403...
10 315 310 0.000048839123899624765...
11 328 53 0.000044915535035844255...
12 391 213 0.000014919239839943413...
13 438 372 0.000008128971060973828...
14 509 234 0.000005265057291740358...
15 659 299 0.000003258910498615529...
16 883 314 0.000002329487103416809...
17 1043 865 0.000000682825287407936...
18 1398 658 0.000000002813997334219...
19 12381 11361 0.000000000872119900887...
20 41956 14509 0.000000000814175918916...
21 50372 7777 0.000000000745870254123...
22 61220 16443 0.000000000305254102745...
23 81580 48414 0.000000000119066473361...
24 189608 5207 0.000000000098904866606...
25 205666 101240 0.000000000041317761770...
26 224305 193798 0.000000000000279730065...
27 4625460 3442529 0.000000000000166375004...
28 4969792 1213224 0.000000000000027648926...
29 8505660 4961227 0.000000000000009234703...
30 12231434 9975776 0.000000000000003807238...
31 20044438 13462483 0.000000000000002464138...
32 27091898 6698814 0.000000000000000000993...
33 807041817 231466141 0.000000000000000000445...
34 2388370134 2050322982 0.000000000000000000225...
35 4024950319 1774903846 0.000000000000000000098...
36 5642521468 4287949990 0.000000000000000000086...
37 5713420381 815914629 0.000000000000000000045...
38 9015369803 7431998815 0.000000000000000000029...
CROSSREFS
KEYWORD
nonn
AUTHOR
Jon E. Schoenfield, Feb 17 2021
EXTENSIONS
Offset corrected by Chai Wah Wu, Mar 03 2021
STATUS
approved