cp's OEIS Frontend

A369659 Non-multiples of 3 whose arithmetic derivative, or equally, the sum of prime factors (with multiplicity) is a multiple of 3.

%I A369659 #47 Jul 08 2024 11:13:00
%S A369659 1,8,14,20,26,35,38,44,50,62,64,65,68,74,77,86,92,95,110,112,116,119,
%T A369659 122,125,134,143,146,155,158,160,161,164,170,185,188,194,196,203,206,
%U A369659 208,209,212,215,218,221,230,236,242,254,275,278,280,284,287,290,299,302,304,305,314,323,326,329,332,335,341,343
%N A369659 Non-multiples of 3 whose arithmetic derivative, or equally, the sum of prime factors (with multiplicity) is a multiple of 3.
%C A369659 This is a subsequence of A373475, containing all its terms that are not multiples of 3. (See comments in A373475 for a proof). The first difference from A373475 is at n=4186, where A373475(4186) = 19683 = 3^9, the value which is missing from this sequence. - _Antti Karttunen_, Jun 07 2024
%C A369659 From _Antti Karttunen_, Jun 11 2024: (Start)
%C A369659 A multiplicative semigroup: if m and n are in the sequence, then so is m*n.
%C A369659 Numbers that are not multiples of 3, and the multiplicities of prime factors of the forms 3m+1 (A002476) and 3m-1 (A003627) are equal modulo 3.
%C A369659 Like A373597, which is a subsequence, also this sequence can be viewed as a kind of k=3 variant of A046337.
%C A369659 A289142, numbers whose sum of prime factors (with multiplicity, A001414) is a multiple of 3, is generated (as a multiplicative semigroup) by the union of this sequence with {3}.
%C A369659 A327863, numbers whose arithmetic derivative is a multiple of 3, is generated by this sequence and A008591.
%C A369659 A373478, numbers that are in the intersection of A289142 and A327863, is generated by the union of this sequence with {9, 27}.
%C A369659 A373475, numbers that are in the intersection of A289142 and A369644 (positions of multiples of 3 in A083345), is generated by the union of this sequence with {19683}, where 19683 = 3^9.
%C A369659 (End)
%C A369659 The integers in the multiplicative subgroup of positive rationals generated by semiprimes of the form 3m+2 (A344872) and cubes of primes except 27. - _Peter Munn_, Jun 19 2024
%H A369659 Antti Karttunen, <a href="/A369659/b369659.txt">Table of n, a(n) for n = 1..33911</a>
%H A369659 <a href="/index/Se#sequences_which_agree_for_a_long_time">Index entries for sequences which agree for a long time but are different</a>
%e A369659 280 = 2*2*2*5*7 is included as it is not a multiple of 3, and one of its prime factors (7) is of the form 3m+1 and four are of the form 3m-1, and because 4 == 1 (mod 3). Also, A001414(280) = 18, and A003415(280) = 516, both of which are multiples of 3. - _Antti Karttunen_, Jun 12 2024
%o A369659 (PARI) \\ See A369658.
%Y A369659 Cf. A001414, A002476, A003415, A003627, A083345, A369658 (characteristic function).
%Y A369659 Intersection of A001651 and A327863.
%Y A369659 Intersection of A001651 and A373475.
%Y A369659 Setwise difference A373475 \ A373476.
%Y A369659 Subsequence of A369644, which is a subsequence of A327863, and also of the following sequences: A289142, A373475, A373478.
%Y A369659 Includes A030078 \ {27}, A344872 and A373597 as subsequences.
%Y A369659 Cf. also A046337, A360110, A369969 for cases k=2, 4, 5 of "Nonmultiples of k whose arithmetic derivative is a multiple of k".
%Y A369659 Cf. also A374044.
%K A369659 nonn
%O A369659 1,2
%A A369659 _Antti Karttunen_, Feb 10 2024
%E A369659 Name amended with an alternative definition by _Antti Karttunen_, Jun 11 2024