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A327863 Numbers whose arithmetic derivative is a multiple of 3, cf. A003415.

%I A327863 #28 Jun 12 2024 16:16:57
%S A327863 0,1,8,9,14,18,20,26,27,35,36,38,44,45,50,54,62,63,64,65,68,72,74,77,
%T A327863 81,86,90,92,95,99,108,110,112,116,117,119,122,125,126,134,135,143,
%U A327863 144,146,153,155,158,160,161,162,164,170,171,180,185,188,189,194,196,198,203,206,207,208,209,212,215,216,218,221,225
%N A327863 Numbers whose arithmetic derivative is a multiple of 3, cf. A003415.
%C A327863 From _Antti Karttunen_, May 27 2024 and Jun 12 2024: (Start)
%C A327863 This is a multiplicative semigroup: if m and n are in the sequence then so is m*n, and is generated by A008591 and A369659.
%C A327863 Term is present if and only if it is either a multiple of 9, or it is not a multiple of 3 and the sum of its prime factors (with repetition, A001414) is a multiple of 3, which happens iff the multiplicities of prime factors of the form 3m+1 (A002476) and of the form 3m-1 (A003627) are equal modulo 3.
%C A327863 (End)
%H A327863 Antti Karttunen, <a href="/A327863/b327863.txt">Table of n, a(n) for n = 1..10000</a>
%o A327863 (PARI)
%o A327863 A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
%o A327863 A359430(n) = !(A003415(n)%3);
%o A327863 isA327863 = A359430;
%Y A327863 Cf. A001414, A002476, A003415, A003627, A235992, A289142, A327862, A327864, A327865, A359430 (characteristic function).
%Y A327863 Positions of 0's in A373253.
%Y A327863 Nonnegative integers are partitioned between this sequence, A373255, and A373257.
%Y A327863 Disjoint union of A008591 and A369659.
%Y A327863 Other subsequences: A327933, A369644, A370119, A373144, A373478, A373494, A373597.
%Y A327863 Cf. also A369654, A370123.
%K A327863 nonn
%O A327863 1,3
%A A327863 _Antti Karttunen_, Sep 30 2019