A347891 -id:A347891 - cp's OEIS frontend

A347890 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.

245025, 540225, 893025, 2205225, 3080025, 4862025, 6125625, 6890625, 7868025, 10989225, 13505625, 14402025, 19847025, 22896225, 23474025, 26471025, 27720225, 29648025, 43758225, 45765225, 55130625, 57836025, 60140025, 65367225, 70812225, 72335025, 76475025, 77000625, 94770225, 121550625, 153140625, 156125025

Offset: 1

Antti Karttunen, Sep 19 2021

nonn

Odd numbers k such that A033880(k) is positive but A342926(k) is negative.
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.
The first term that is not a multiple of 25 is a(146) = 6800806089 = 82467^2.
This is not a subsequence of A325311. The first term that is not present there is a(5) = 3080025.

Index entries for sequences related to sigma(n)

Intersection of A005231 and A343216.
Subsequence of A016754, of A156942 and of A347889 (its odd terms).
Cf. A000203, A003415, A033880, A325311, A342926, A347891 (the square roots).

\\ Using the program given in A347891 would be much faster than this:
A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
isA347890(n) = ((n%2)&&(A003415(sigma(n))(2*n)));

cp's OEIS Frontend

A347890 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.

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