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A369252 Arithmetic derivative applied to the numbers of the form p*q*r where p,q,r are (not necessarily distinct) odd primes.

%I A369252 #26 Jan 24 2024 16:44:23
%S A369252 27,39,51,55,75,71,87,75,91,111,103,123,95,119,147,131,119,151,183,
%T A369252 151,135,195,167,155,231,147,199,191,187,255,167,267,211,291,195,215,
%U A369252 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 A369252 Arithmetic derivative applied to the numbers of the form p*q*r where p,q,r are (not necessarily distinct) odd primes.
%C A369252 The table showing the possible modulo 3 combinations for p, q, r and the sum ((p*q) + (p*r) + (q*r)):
%C A369252   |   p  |   q  |   r  | sum ((p*q) + (p*r) + (q*r)) (mod 3)
%C A369252 --+------+------+------+----------------------------------------
%C A369252   |   0  |   0  |   0  |  0, p=q=r=3, sum is 27.
%C A369252 --+------+------+------+----------------------------------------
%C A369252   |   0  |   0  | +/-1 |  0, p=q=3, r > 3.
%C A369252 --+------+------+------+----------------------------------------
%C A369252   |   0  |  +1  |  +1  | +1
%C A369252 --+------+------+------+----------------------------------------
%C A369252   |   0  |  -1  |  -1  | +1
%C A369252 --+------+------+------+----------------------------------------
%C A369252   |   0  |  -1  |  +1  | -1
%C A369252 --+------+------+------+----------------------------------------
%C A369252   |   0  |  +1  |  -1  | -1
%C A369252 --+------+------+------+----------------------------------------
%C A369252   |  +1  |  +1  |  +1  |  0
%C A369252 --+------+------+------+----------------------------------------
%C A369252   |  -1  |  -1  |  -1  |  0
%C A369252 --+------+------+------+----------------------------------------
%C A369252   |  -1  |  +1  |  +1  | -1, regardless of the order, thus x3.
%C A369252 --+------+------+------+----------------------------------------
%C A369252   |  +1  |  -1  |  -1  | -1, regardless of the order, thus x3.
%C A369252 --+------+------+------+----------------------------------------
%C A369252 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 A369252 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 A369252           6,         2,         1,
%C A369252          39,        22,        38,
%C A369252         291,       209,       499,
%C A369252        2527,      1884,      5588,
%C A369252       23527,     17020,     59452,
%C A369252      227297,    156240,    616462,
%C A369252     2232681,   1453030,   6314288,
%C A369252    22119496,  13661893,  64218610,
%C A369252   220098425, 129624002, 650277572.
%C A369252 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 A369252 Antti Karttunen, <a href="/A369252/b369252.txt">Table of n, a(n) for n = 1..20000</a>
%F A369252 a(n) = A003415(A046316(n)).
%Y A369252 Cf. A369251 (same sequence sorted into ascending order, with duplicates removed).
%Y A369252 Cf. A369464 (numbers that do not occur in this sequence).
%Y A369252 Cf. A003415, A046316, A369054, A369058, A369248.
%Y A369252 Cf. also the trisections of A369055: A369460, A369461, A369462 and their partial sums A369450, A369451, A369452, also A369241, A369245.
%Y A369252 Only terms of A004767 occur here.
%K A369252 nonn
%O A369252 1,1
%A A369252 _Antti Karttunen_, Jan 22 2024