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

1,1

Comments

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.

Links

Table of n, a(n) for n=1..32.

Index entries for sequences related to sigma(n)

Prog

(PARI)
\\ 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))<n)&&(sigma(n)>(2*n)));

Crossrefs

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).

Sequence in context: A186801 A157761 A349086 * A187771 A233632 A251856

Adjacent sequences: A347887 A347888 A347889 * A347891 A347892 A347893

Keyword

nonn

Author

Antti Karttunen, Sep 19 2021

Status

approved