A341053 Irregular triangle read by rows: T(n, k) is the k-th largest numerator of the A341052(n) ratios for which there exist A341051(n) n-digit integers (the maximum possible) that are in geometric progression.

2, 3, 3, 4, 4, 5, 5, 5, 5, 5, 6, 7, 6, 7, 8, 7, 7, 7, 8, 7, 8, 9, 8, 9, 8, 9, 8, 9, 9, 9, 9, 10, 11, 10, 11, 11, 11, 11, 11, 11, 12, 12, 13, 12, 13, 13, 14, 13, 14, 13, 14, 15, 14, 14, 14, 15, 15, 14, 15, 16, 15, 15, 16, 17, 15, 16, 17, 16, 16, 16, 17, 18, 17

Offset

1,1

Comments

The ratios are of the form m/(m-1) with m > 1.

The first 4 ratios are 2, 3/2, 3/2, 4/3, then, A341053(5) = 2 with the two possible ratios 4/3 and 5/4, then the successive ratios are 5/4, 5/4, 5/4 and A341053(9) = 3 with these 3 possible ratios 5/4, 6/5, 7/6.

The corresponding denominator of these A341052(n) ratios is equal to T(n, k) - 1.

Links

Table of n, a(n) for n=1..73.

Diophante, A10219, Progressions maximales (in French).

Bernard Schott, tabf for A341053.

Example

There exist 6 integers in the largest possible string with 3-digit numbers that are in geometric progression (128, 192, 288, 432, 648, 972), and this string is obtained with the ratio = 3/2, so T(3, 1) = 3.
There exist 8 integers in the largest possible string with 5-digit numbers that are in geometric progression, and two such strings are obtained with these 2 distinct following ratios:
-> with ratio = 4/3, the 8 integers go from 10935 to 81920,
-> with ratio = 5/4, the 8 integers go from 16384 to 78125.
so T(5, 1) = 4, T(5, 2) = 5.
Triangle begins:
  2;
  3;
  3;
  4;
  4, 5;
  5;
  5;
  5;
  5, 6, 7;
  ...

Crossrefs

Cf. A341051, A341052.

Sequence in context: A320757 A061716 A362692 * A126236 A198194 A073047

Adjacent sequences: A341050 A341051 A341052 * A341054 A341055 A341056

Keyword

nonn,base,tabf

Author

Bernard Schott, Apr 20 2021

Extensions

More terms from Jinyuan Wang, Apr 23 2021

Status

approved