cp's OEIS Frontend

A373475 Numbers k such that A001414(k) and A083345(k) are both multiples of 3, where A001414 is fully additive with a(p) = p, and A083345 is the numerator of the fully additive function with a(p) = 1/p.

%I A373475 #26 Jun 07 2024 14:50:02
%S A373475 1,8,14,20,26,35,38,44,50,62,64,65,68,74,77,86,92,95,110,112,116,119,
%T A373475 122,125,134,143,146,155,158,160,161,164,170,185,188,194,196,203,206,
%U A373475 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 A373475 Numbers k such that A001414(k) and A083345(k) are both multiples of 3, where A001414 is fully additive with a(p) = p, and A083345 is the numerator of the fully additive function with a(p) = 1/p.
%C A373475 If k is a term, then 3^9 * k is also a term. See A373476.
%C A373475 A369659 is a subsequence of this sequence, giving the terms that are not multiples of 3. This follows because A083345(n) = n' / gcd(n',n) and from the following lemma: When k is not a multiple of 3, then either sopfr(k) [= A001414(k)] and k' [= A003415(k)] are both multiples of 3, or both are non-multiples of 3.
%C A373475 Proof of the lemma: As k is not a multiple of 3, all its prime factors p, q, r, s, t, u, v, w, ... (not necessarily all distinct) are either of the form 3m+1 or 3m-1. Let's first eliminate from k all triplets of primes that are of the same type modulo 3, either -1 or +1, (marked now as p, q, r) as they do not affect the divisibility by 3 of either the sopfr(k) or k'. In the case of the arithmetic derivative this is because we have k' = (pqr)' * (k/pqr) + (k/pqr)' * pqr, and as we know that the first summand is a multiple of 3 (because (pqr)' is), therefore the divisibility of the whole expression by 3 depends only on whether (k/pqr)' is a multiple of 3, as certainly pqr is not a multiple of 3.
%C A373475   What will remain after such elimination process has been completed as far as possible, must be either 1, or of the form p*q (p and q of different types), or p*q*r*s (with two primes of one type, and two primes of the other type), in which cases both sopfr(k) and k' are multiples of 3, or then alternatively, what remains must be of the form p*q (p and q of the same type), or p*q*r (with two primes of one type and the third of the other type), both cases which indicate that both sopfr(k) and k' are non-multiples of 3.
%H A373475 Antti Karttunen, <a href="/A373475/b373475.txt">Table of n, a(n) for n = 1..33911</a>
%H A373475 <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>
%F A373475 a(n) = A373476(n) / 3^9.
%e A373475 110 = 2*5*11 is a term of this sequence because 2+5+11 = 18 is a multiple of 3, and also 2*5 + 2*11 + 5*11 = 87 is a multiple of 3.
%e A373475 54 (= A369644(10)) is NOT a term of this sequence, because A001414(54) = 11 is not a multiple of 3, although A083345(54) = 3 is.
%e A373475 19683 = 3^9 is a term of this sequence, because both A001414(19683) = 9*3 = 27 and A083345(19683) = A003415(3^9)/gcd(3^9, A003415(3^9)) = 3, are multiples of 3.
%o A373475 (PARI) isA373475 = A373474;
%Y A373475 Cf. A001414, A003415, A083345, A373474 (characteristic function).
%Y A373475 Positions of multiples of 3 in A373363.
%Y A373475 Intersection of A289142 and A369644.
%Y A373475 Subsequence of A373478.
%Y A373475 Disjoint union of A369659 and A373476.
%Y A373475 Differs from A369659 for the first time at n=4186, where a(4186) = A373476(1) = 19683, a term not present in A369659, as it is the first multiple of 3 in this sequence.
%K A373475 nonn
%O A373475 1,2
%A A373475 _Antti Karttunen_, Jun 06 2024