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A373599 Numbers k such that k and A327860(k) are both multiples of 3, where A327860 is the arithmetic derivative of the primorial base exp-function.

%I A373599 #19 Jun 18 2024 16:19:17
%S A373599 0,18,36,54,72,90,108,126,144,162,180,198,222,240,258,276,294,312,330,
%T A373599 348,366,384,402,426,444,462,480,498,516,534,552,570,588,606,624,630,
%U A373599 648,666,684,702,720,738,756,774,792,810,828,852,870,888,906,924,942,960,978,996,1014,1032,1056,1074,1092,1110,1128
%N A373599 Numbers k such that k and A327860(k) are both multiples of 3, where A327860 is the arithmetic derivative of the primorial base exp-function.
%C A373599 If x and y are terms and if A329041(x,y) = 1 (i.e., when adding x and y together will not generate any carries in the primorial base), then x+y is also a term. This follows from the quasi-exponential nature of A276086 and because A373144 is a multiplicative semigroup.
%H A373599 Antti Karttunen, <a href="/A373599/b373599.txt">Table of n, a(n) for n = 1..20020</a>
%H A373599 <a href="/index/Pri#primorialbase">Index entries for sequences related to primorial base</a>
%e A373599 18 = 3*6 is included, because also A327860(18) = 75 is a multiple of 3.
%e A373599 222 = 3*74 is included, because also A327860(222) = 135 is a multiple of 3.
%e A373599 240 = 3*80 is included, because also A327860(240) = 18 is a multiple of 3.
%e A373599 258 = 3*86 is included, because also A327860(258) = 8025 is a multiple of 3. Note that A049345(18) = 300, A049345(240) = 11000, and A049345(240+18) = 11300, so the sum in this case is carry-free (cf. the comment).
%e A373599 2556 = 3*852 is included, because also A327860(2556) = 2556 is a multiple of 3 (see also A328110 and A373144).
%o A373599 (PARI) isA373599 = A373598;
%Y A373599 Cf. A049345, A276086, A327860, A329041, A373598 (characteristic function).
%Y A373599 Indices of multiples of 3 in A351083.
%Y A373599 Intersection of A008585 and A369654.
%Y A373599 Differs from A008600 (multiples of 18) for the first time at a(13) = 222, which is not a multiple of 18.
%Y A373599 Cf. also A373144.
%K A373599 nonn
%O A373599 1,2
%A A373599 _Antti Karttunen_, Jun 18 2024