>>7695

Don't sweat it.

I've been spending some more time looking into the smooth numbers. I've noticed that given (e, n) the smooth numbers for those records (bigN - n) exists in (e - 2*n, 1). If e is odd, then the smooth numbers will exist as d's. If e is even then smooth number will appear as a. If you jump to (e + 2n, n) the smooth values of those records will match the a's or d's in (e, 1) at t for bigN. Essentially, given a record a, b we get bigN. If e is odd then the d at a[t] = aBigN (e, 1) will be a "smooth" number for the jump (or a + 1, b + 1) in (e + 2n, n).

The smooth numbers of (0, 1) exist in (-2, 1) at t = 1, 4, 9, 16…

The smooth numbers of (1, 1) exist in (-1, 1) at t = 1, 4, 9, 16…

It also appears that the t's for the smooth numbers in (e, 1) exist in (e - 1, 2). Although that is just a preliminary result. I haven't had much time to look into that result.

>>7704

The patterns aren't always [1, 4, 9, ..] but offsets.

Take (7, 1). The first 10 smooth numbers are:

[2, 5, 10, 17, 26, 37, 50, 65, 82]

Here the relationship with squares is 2 + 2, 5 + 4, 10+6, 17 + 8.

(8, 1) has first 10 smooth numbers: [4, 8, 14, 22, 32, 44, 58, 74, 92] which has the pattern 4 + 0, 8 + 1, 14 + 2, 22 + 3, 32 + 4 …

>>7705

Take cell (3, 6). Moving to the left will put us in (3 + 2*6, 6) =(15, 6).

The first 10 rows in (3, 6) is:

and their smooth numbers are:

{3:6:4:3:1:19}

{3:6:16:9:7:37}

{3:6:34:15:19:61}

{3:6:58:21:37:91}

{3:6:88:27:61:127}

{3:6:124:33:91:169}

{3:6:166:39:127:217}

{3:6:214:45:169:271}

{3:6:268:51:217:331}

{3:6:328:57:271:397}

[0, 108, 540, 1620, 3780, 7560, 13608, 22680, 35640]

While the first 10 rows in (15, 6) are:

{15:6:17:9:8:38}

{15:6:35:15:20:62}

{15:6:59:21:38:92}

{15:6:89:27:62:128}

{15:6:125:33:92:170}

{15:6:167:39:128:218}

{15:6:215:45:170:272}

{15:6:269:51:218:332}

{15:6:329:57:272:398}

{15:6:395:63:332:470}

and their smooth numbers are:

[129, 579, 1683, 3873, 7689, 13779, 22899, 35913, 53793]

Note that (15, 6) has "shifted" one record down, meaning the first cell in (3, 6) (a=1, b=19) is gone.

Each of the smooth numbers in (15, 6) are the d's at (3, 1) where a[t] = BigN.

The big n for a=7, b=37 is 114 and the d at that location = 129 and exists at t = 8.

This means that the t's in (3, 1) where d = smooth number for (15, 6) are: [8, 17, 29, 44, 62, 83, 107, 134, 164]

>>7706

Minor typo, pasted the records along with the smooth numbers, but you'll see the difference.

The smooth numbers for (3, 6) (0, 108, …) exists in (3 - 2*6, 1) =(-9, 1) at t's: [2, 8, 17, 29, 44, 62, 83, 107, 134]

Which we can see occur at the same location in (3, 1) as the smooth numbers do for (15, 6).

>>7709

Hey man, that's understandable. You need to take care of yourself. You've done a massive job and I'm very impressed (and a bit guilty for not participating as much in the pattern thread). I hope you'll straighten things out.