|
%I #19 Jan 23 2024 16:29:20
%S 399,4809,5763,63021,76449,1301673,19204051701,421177029231,
%T 908999759928891,39248269334566041,39248273246018313,
%U 68437232802099093891,4903038892893242229501
%N 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.
%C All terms are multiples of 3 because for all n >= 2, A006862(n) = 1 + A002110(n) == +1 (mod 3), see A369252.
%C Although the first thirteen terms appear in the ascending order, this might not be true for all the later terms, if they exist.
%e Rows 1..3 have no terms.
%e Row 4 has one term: 399 = 3 * 7 * 19, whose arithmetic derivative (see A003415) 399' is 211 = 1 + prime(4)# [= A006862(4)].
%e Row 5 has two terms: 4809 = 3 * 7 * 229 and 5763 = 3 * 17 * 113, with 4809' = 5763' = 2311 = 1 + prime(5)#.
%e Row 6 has two terms: 63021 = 3 * 7 * 3001 and 76449 = 3 * 17 * 1499, with 63021' = 76449' = 30031 = 1 + prime(6)#.
%e Row 7 has one term: 1301673 = 3 * 17 * 25523, whose arithmetic derivative is 510511 = 1 + prime(7)#.
%e Rows 8 and 9 have no terms.
%e Row 10 has one term: 19204051701 = 3 * 281 * 22780607, whose arithmetic derivative is 6469693231 = 1 + prime(10)#.
%e Row 11 has one term: 421177029231 = 3 * 7 * 20056049011, whose arithmetic derivative is 200560490131 = 1 + prime(11)#.
%e Row 12 has no terms.
%e Row 13 has one term: 908999759928891 = 3 * 727 * 416781182911, whose arithmetic derivative is 304250263527211 = 1 + prime(13)#.
%e Row 14 has two terms: 39248269334566041 = 3 * 8071457 * 1620866771, and 39248273246018313 = 3 * 11056387 * 1183276033, which both have arithmetic derivative 13082761331670031 = 1 + prime(14)#.
%e Row 15 has no terms.
%e Row 16 has one term: 68437232802099093891 = 3 * 7 * 3258915847719004471, whose arithmetic derivative is 32589158477190044731 = 1 + prime(16)#.
%e Row 17 has at least this term: 4903038892893242229501 = 3 * 17 * 96138017507710631951, whose arithmetic derivative is 1922760350154212639071 = 1 + prime(17)#.
%Y Subsequence of A008585 and of A046316.
%Y Cf. A003415, A002110, A006862, A369054, A369245 (counts of solutions, length of row n), A369252.
%Y Cf. also A366890, A369240 for similar tables.
%K nonn,tabf,hard,more
%O 1,1
%A _Antti Karttunen_, Jan 22 2024
| 