[seqfan] Stern's diatomic series, n-phi-torial and the primes

[Tilman Piesk]

Hi all.
Recently I found an interesting feature of Stern's diatomic series, that
seems to deserve further investigation:

In the rows of A002487 seen as a triangle some entries appear most often.
E.g. in row 5 the 7 appears most often (four times). In row 6 the 9 and the
11 are most often (also four times).

I collected these most numerous entries (sequence MNE for now),
and found that ordered they are similar to A193339, the indices of record
values in n-phi-torial.
All numbers in MNE I found are in A193339.
And when after collecting the MNE until row 9 the 37 between 31 and 41 is
missing - voila: row 10 brings it (along with 43, the next entry of
A193339).
So MNE is probably a subsequence of A193339, and chances are that they are
equal.

It must be mentioned, that probably, after the first few numbers, A193339
continues like the prime numbers.

The following table shows, which row brings which new MNE:

.row.|...... new MNE ........|.indices of A193339
_____|_______________________|___________________
.1...|.....1.................|.....1
.2
.3...|.....3.................|.....2
.4...|.....4....5............|.....3..4
.5...|.....7.................|.....5
.6...|.....9...11............|.....6..7
.7
.8...|....13...17............|.....8..9
.9...|....19...23...31...41..|....10..11..13..15
10...|....37...43............|....14..16
11...|....71.................|....22
12
13...|...127.................|....33
14...|...109.................|....31
15...|...199..251............|....48..56
16...|...263.................|....58
17...|...433.................|....86
18...|...701.................|...128
19...|...839.................|...148
20...|..1193.................|...198
21...|..1801.................|...281
22...|..2693.................|...394
23...|..4229.................|...581
24...|..4349.................|...596
25...|..7759.................|...987
26...|.11287.................|..1367
27...|.14627.................|..1714
28...|.20929.................|..2356

The index numbers in the (ruins of the) right column are correct under the
assumption, that A193339 continues like the primes.

I hope someone more knowledgeable will look into this.

By the way: Also my nimber related conjecture in A002487 could need proof
or disproof.

Greetings,
Tilman Piesk alias mate2code