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Numbers n such that sigma(2n) < sigma(2n+1).
2

%I #19 Apr 25 2019 03:32:20

%S 1,31,37,67,73,97,127,157,199,202,229,241,247,262,277,283,307,313,331,

%T 337,346,367,379,382,397,409,427,457,472,487,499,517,547,562,577,607,

%U 619,643,661,697,727,757,769,787,823,829,841,877,892,907,913,922,937

%N Numbers n such that sigma(2n) < sigma(2n+1).

%C There are 2301 primes and 3169 composites among the 5470 first terms. Does limit n->infinity card(k : a(k) prime)/card(k : a(k) composite) > 0 ?

%H Vaclav Kotesovec, <a href="/A082957/b082957.txt">Table of n, a(n) for n = 1..10000</a>

%H Mits Kobayashi, Tim Trudgian, <a href="https://arxiv.org/abs/1904.10064">On integers n for which sigma(2n+1)>=sigma(2n)</a>, arXiv:1904.10064 [math.NT], 2019.

%H Vaclav Kotesovec, <a href="/A082957/a082957.jpg">Plot of a(n)/n for n = 1..50000</a>

%F Conjecture : a(n) is asymptotic to c*n where 18<c<18.5.

%p q:= n-> (s-> s(2*n)<s(2*n+1))(numtheory[sigma]):

%p select(q, [$1..1000])[]; # _Alois P. Heinz_, Apr 24 2019

%t Select[Range[1, 1000], DivisorSigma[1,2*#] < DivisorSigma[1,2*#+1]&] (* _Vaclav Kotesovec_, Feb 15 2019 *)

%o (PARI) isok(n) = sigma(2*n) < sigma(2*n+1); \\ _Michel Marcus_, Dec 04 2013

%K nonn

%O 1,2

%A _Benoit Cloitre_, May 26 2003