A369246 Irregular triangle read by rows, where row n lists in ascending order all numbers k in A046316 for which k' = the n-th Euclid number, where k' stands for the arithmetic derivative, and the Euclid numbers are given by A006862. Rows of length zero are simply omitted, i.e., when A369245(n) = 0.

399, 4809, 5763, 63021, 76449, 1301673, 19204051701, 421177029231, 908999759928891, 39248269334566041, 39248273246018313, 68437232802099093891, 4903038892893242229501

Offset: 1

Author

Antti Karttunen, Jan 22 2024

Keywords

• nonn
• tabf
• hard
• more

Comments

All terms are multiples of 3 because for all n >= 2, A006862(n) = 1 + A002110(n) == +1 (mod 3), see A369252.

Although the first thirteen terms appear in the ascending order, this might not be true for all the later terms, if they exist.

Examples

			Rows 1..3 have no terms.
Row 4 has one term: 399 = 3 * 7 * 19, whose arithmetic derivative (see A003415) 399' is 211 = 1 + prime(4)# [= A006862(4)].
Row 5 has two terms: 4809 = 3 * 7 * 229 and 5763 = 3 * 17 * 113, with 4809' = 5763' = 2311 = 1 + prime(5)#.
Row 6 has two terms: 63021 = 3 * 7 * 3001 and 76449 = 3 * 17 * 1499, with 63021' = 76449' = 30031 = 1 + prime(6)#.
Row 7 has one term: 1301673 = 3 * 17 * 25523, whose arithmetic derivative is 510511 = 1 + prime(7)#.
Rows 8 and 9 have no terms.
Row 10 has one term: 19204051701 = 3 * 281 * 22780607, whose arithmetic derivative is 6469693231 = 1 + prime(10)#.
Row 11 has one term: 421177029231 = 3 * 7 * 20056049011, whose arithmetic derivative is 200560490131 = 1 + prime(11)#.
Row 12 has no terms.
Row 13 has one term: 908999759928891 = 3 * 727 * 416781182911, whose arithmetic derivative is 304250263527211 = 1 + prime(13)#.
Row 14 has two terms: 39248269334566041 = 3 * 8071457 * 1620866771, and 39248273246018313 = 3 * 11056387 * 1183276033, which both have arithmetic derivative 13082761331670031 = 1 + prime(14)#.
Row 15 has no terms.
Row 16 has one term: 68437232802099093891 = 3 * 7 * 3258915847719004471, whose arithmetic derivative is 32589158477190044731 = 1 + prime(16)#.
Row 17 has at least this term: 4903038892893242229501 = 3 * 17 * 96138017507710631951, whose arithmetic derivative is 1922760350154212639071 = 1 + prime(17)#.
		

Crossrefs

Subsequence of A008585 and of A046316.

Cf. A003415, A002110, A006862, A369054, A369245 (counts of solutions, length of row n), A369252.

Cf. also A366890, A369240 for similar tables.