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A358395 Odd numbers k such that sigma(k) + sigma(k+2) > 2*sigma(k+1); odd terms in A053228.

%I A358395 #17 Nov 17 2022 14:12:20
%S A358395 1125,1573,1953,2205,2385,3465,5185,5353,5773,6433,6613,6825,7245,
%T A358395 7425,7665,7693,8505,8925,9133,9205,9405,9945,10393,10773,11473,11653,
%U A358395 12285,12493,12705,13473,13585,13725,14025,15013,15145,15433,16065,16245,16905,17253,17325,17953
%N A358395 Odd numbers k such that sigma(k) + sigma(k+2) > 2*sigma(k+1); odd terms in A053228.
%C A358395 Odd numbers k such that A053223(k) > 0.
%C A358395 Terms congruent to 5 modulo 6 exist but must be very large: for example A053223(670173643268502741420822977335461337017377351999597045900203591953125) = 1311786588705365455963902347308393766941056366825184647502989937872.
%C A358395 A number m coprime to 2 and 3 such that sigma(m)/m >= 3 (m = A358412(3) = A358413(2) = 5^4*7^3*11^2*13^2*17*...*157 ~ 5.16404*10^66 is the smallest such number; see the link from Mercurial, the Spectre) produces a family of infinitely many terms congruent to 5 modulo 6 in this sequence, by Dirichlet's theorem on arithmetic progressions. Concretely, let k == 5 (mod 6), N(t) = t*k*(k+2) + (k+1)/6 for t >= 0, then:
%C A358395 (i) If sigma(k)/k >= 3. If N(t) is prime and 6*N(t)+1 is composite, then sigma(6*N(t)-1) >= 3*(6*N(t)-1), sigma(6*N(t)) = 12*(N(t)+1) and sigma(6*N(t)+1) >= 1+sqrt(6*N(t)+1)+(6*N(t)+1), so A053223(6*N(t)-1) >= sqrt(6*N(t)+1) - 25 >= sqrt(k+2) - 25 > 0.
%C A358395 (ii) If sigma(k+2)/(k+2) >= 3. If N(t) is prime and 6*N(t)-1 is composite, then sigma(6*N(t)+1) >= 3*(6*N(t)+1), sigma(6*N(t)) = 12*(N(t)+1) and sigma(6*N(t)-1) >= 1+sqrt(6*N(t)-1)+(6*N(t)-1), so A053223(6*N(t)-1) >= sqrt(6*N(t)-1) - 21 >= sqrt(k) - 21 > 0.
%H A358395 Jianing Song, <a href="/A358395/b358395.txt">Table of n, a(n) for n = 1..20000</a>
%H A358395 Mercurial, the Spectre, <a href="http://hi.gher.space/forum/viewtopic.php?f=11&amp;t=2248&amp;sid=cbf9e6743a4ccdcd6cbcadcdf56946db">Abundant numbers coprime to n</a>, Hi.gher. Space.
%H A358395 Wikipedia, <a href="http://en.wikipedia.org/wiki/Dirichlet%27s_theorem_on_arithmetic_progressions">Dirichlet's theorem on arithmetic progressions</a>
%e A358395 1125 is a term since sigma(1126) = 1692 is smaller than the average of sigma(1125) = 2028 and sigma(1127) = 1368.
%o A358395 (PARI) isA358395(n) = (n%2) && (sigma(n) + sigma(n+2) > 2*sigma(n+1))
%Y A358395 Cf. A053228, A053223, A000203 (sigma), A358396.
%Y A358395 Cf. also A358412, A358413.
%K A358395 nonn
%O A358395 1,1
%A A358395 _Jianing Song_, Nov 13 2022