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%I #19 Sep 19 2021 22:02:53
%S 245025,540225,893025,2205225,3080025,4862025,6125625,6890625,7868025,
%T 10989225,13505625,14402025,19847025,22896225,23474025,26471025,
%U 27720225,29648025,43758225,45765225,55130625,57836025,60140025,65367225,70812225,72335025,76475025,77000625,94770225,121550625,153140625,156125025
%N Odd numbers k such that sigma(k) > 2*k and A003415(sigma(k)) < k, where A003415 is the arithmetic derivative, and sigma is the sum of divisors function.
%C Odd numbers k such that A033880(k) is positive but A342926(k) is negative.
%C This is a subsequence of A156942, "odd abundant numbers whose abundance is odd". Proof: If sigma(k) > 2*k, and sigma(k) were even, then sigma(k)/2 would be an integer and a divisor of sigma(k), and we could compute A003415(sigma(k)) as A003415(2)*(sigma(k)/2) + 2*A003415(sigma(k)/2) by the definition of the arithmetic derivative. But that value is certainly larger than k, because sigma(k)/2 > k, therefore sigma(k) must be an odd number, with also its abundance sigma(k)-(2k) odd. This also entails that all terms are squares. See A347891 for the square roots.
%C The first term that is not a multiple of 25 is a(146) = 6800806089 = 82467^2.
%C This is not a subsequence of A325311. The first term that is not present there is a(5) = 3080025.
%H <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>
%o (PARI)
%o \\ Using the program given in A347891 would be much faster than this:
%o A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
%o isA347890(n) = ((n%2)&&(A003415(sigma(n))<n)&&(sigma(n)>(2*n)));
%Y Intersection of A005231 and A343216.
%Y Subsequence of A016754, of A156942 and of A347889 (its odd terms).
%Y Cf. A000203, A003415, A033880, A325311, A342926, A347891 (the square roots).
%K nonn
%O 1,1
%A _Antti Karttunen_, Sep 19 2021
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