%I #26 Jan 24 2024 16:44:23

%S 27,39,51,55,75,71,87,75,91,111,103,123,95,119,147,131,119,151,183,

%T 151,135,195,167,155,231,147,199,191,187,255,167,267,211,291,195,215,

%U 247,191,263,215,327,251,247,363,203,375,311,271,255,239,411,231,311,343,299,231,435,359,331,447,311,263,391,483,263

%N Arithmetic derivative applied to the numbers of the form p*q*r where p,q,r are (not necessarily distinct) odd primes.

%C The table showing the possible modulo 3 combinations for p, q, r and the sum ((p*q) + (p*r) + (q*r)):

%C | p | q | r | sum ((p*q) + (p*r) + (q*r)) (mod 3)

%C --+------+------+------+----------------------------------------

%C | 0 | 0 | 0 | 0, p=q=r=3, sum is 27.

%C --+------+------+------+----------------------------------------

%C | 0 | 0 | +/-1 | 0, p=q=3, r > 3.

%C --+------+------+------+----------------------------------------

%C | 0 | +1 | +1 | +1

%C --+------+------+------+----------------------------------------

%C | 0 | -1 | -1 | +1

%C --+------+------+------+----------------------------------------

%C | 0 | -1 | +1 | -1

%C --+------+------+------+----------------------------------------

%C | 0 | +1 | -1 | -1

%C --+------+------+------+----------------------------------------

%C | +1 | +1 | +1 | 0

%C --+------+------+------+----------------------------------------

%C | -1 | -1 | -1 | 0

%C --+------+------+------+----------------------------------------

%C | -1 | +1 | +1 | -1, regardless of the order, thus x3.

%C --+------+------+------+----------------------------------------

%C | +1 | -1 | -1 | -1, regardless of the order, thus x3.

%C --+------+------+------+----------------------------------------

%C Notably a(n) is a multiple of 3 only when A046316(n) is either a multiple of 9, or all primes p, q and r are either == +1 (mod 3) or all are == -1 (mod 3), and the case a(n) == +1 (mod 3) is only possible when A046316(n) is a multiple of 3, but not of 9, and furthermore, it is required that r == q (mod 3). See how these combinations affects sequences like A369241, A369245, A369450, A369451, A369452.

%C For n=1..9 the number of terms of the form 3k, 3k+1 and 3k+2 in range [1..10^n-1] are:

%C 6, 2, 1,

%C 39, 22, 38,

%C 291, 209, 499,

%C 2527, 1884, 5588,

%C 23527, 17020, 59452,

%C 227297, 156240, 616462,

%C 2232681, 1453030, 6314288,

%C 22119496, 13661893, 64218610,

%C 220098425, 129624002, 650277572.

%C It seems that 3k+2 terms are slowly gaining at the expense of 3k+1 terms when n grows, while the density of the multiples of 3 might converge towards a limit.

%H Antti Karttunen, <a href="/A369252/b369252.txt">Table of n, a(n) for n = 1..20000</a>

%F a(n) = A003415(A046316(n)).

%Y Cf. A369251 (same sequence sorted into ascending order, with duplicates removed).

%Y Cf. A369464 (numbers that do not occur in this sequence).

%Y Cf. A003415, A046316, A369054, A369058, A369248.

%Y Cf. also the trisections of A369055: A369460, A369461, A369462 and their partial sums A369450, A369451, A369452, also A369241, A369245.

%Y Only terms of A004767 occur here.

%K nonn

%O 1,1

%A _Antti Karttunen_, Jan 22 2024