A322049 When A322050 is displayed as a triangle the rows converge to this sequence.

1, 7, 6, 30, 8, 48, 17, 81, 9, 50, 29, 145, 27, 145, 37, 189, 8, 45, 34, 166, 45, 252, 73, 342, 37, 179, 89, 425, 74, 374, 86, 412, 8, 49, 33, 165, 46, 270, 91, 436, 50, 277, 149, 734, 122, 630, 144, 723, 38, 179, 101, 488, 130, 753, 209, 990, 90, 450, 210, 991

Offset: 0

Author

N. J. A. Sloane, Dec 15 2018

Keywords

• nonn
• look

Comments

It would be nice to have a formula or recurrence. There is certainly a lot of structure.

Indices of records of a(n)/n are (1, 3, 7, 11, 23, 27, 43, 55, 87, 91, 119, 171, 183, 343, 347, 363, 367, 375, 439, 695, 731, 887, 1367, 1371, 1391, 1399, 1451, 1463, 2743, 2923, 2927, 2935, 3511, ...). The ratio a(n)/n increases roughly by 1 at each of these. We conjecture that this ratio is unbounded. We note that the record ratios occur in "clusters" at indices twice as large as the preceding cluster: 87, 91; 171, 183; 343..375; 695..731; 1367..1463; 2743..2935; ... This is compatible with the self-similar structure of the graph of this sequence, which starts over at a(2^k) = 8 for all k >= 4. (But note also the distinctive substructure repeating with period 2^10, cf. the "logarithmic plot" link.) - M. F. Hasler, Dec 18 2018

Links

• Hugo Pfoertner, Table of n, a(n) for n = 0..5461
• Hugo Pfoertner, Logarithmic plot of 5462 terms, use zoom to see details.

Crossrefs

Cf. A319018, A319019, A322048, A322050, A322051.

Formula

From M. F. Hasler, Dec 18 2018: (Start)

Experimental data suggests the following properties:

Sporadic values occurring only a finite number of times, with no regular pattern:

a(n) | 1 | 6 | 7 | 9 | 37 | 48 | 50 | 53 | ...

-----+---+---+---+---+--------+----+-------+----+-----

n | 0 | 2 | 1 | 8 | 14, 24 | 5 | 9, 40 | 80 | ...

Values occurring in regular patterns:

a(n) = 8 iff n = 2^k, k = 2 or k >= 4; a(n) > 8 for all other n > 2.

a(n) = 33 iff n = 2^(2k+1) + 2, k >= 2; a(n) > 33 for all other n > 12 unless n = 2^k <=> a(n) = 8.

a(n) = 34 iff n = 4^k + 2, k >= 2.

a(n) = 38 iff n = 3*2^k, k = 4, 5, 6, 8, 10, ...

a(n) = 27*2^m if n = 3*2^k with k = 2 (m = 0) or k = 7, 9, ... (m = 1, 2, ...)

a(n) = 45 iff n = 20 or n = 4^k + 1, k >= 2.

a(n) = 46 iff n = 2^(2k+1) + 4, k >= 2.

a(n) = 49 iff n = 2^(2k+1) + 1, k >= 2, or n = 4^k + 4, k >= 3.

a(n) > 50 for all n > 10 not mentioned above. (End)