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

1125, 1573, 1953, 2205, 2385, 3465, 5185, 5353, 5773, 6433, 6613, 6825, 7245, 7425, 7665, 7693, 8505, 8925, 9133, 9205, 9405, 9945, 10393, 10773, 11473, 11653, 12285, 12493, 12705, 13473, 13585, 13725, 14025, 15013, 15145, 15433, 16065, 16245, 16905, 17253, 17325, 17953

Offset: 1

Author

Jianing Song, Nov 13 2022

Keywords

• nonn

Comments

Odd numbers k such that A053223(k) > 0.

Terms congruent to 5 modulo 6 exist but must be very large: for example A053223(670173643268502741420822977335461337017377351999597045900203591953125) = 1311786588705365455963902347308393766941056366825184647502989937872.

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:

(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.

(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.

Examples

			1125 is a term since sigma(1126) = 1692 is smaller than the average of sigma(1125) = 2028 and sigma(1127) = 1368.
		

Links

• Jianing Song, Table of n, a(n) for n = 1..20000
• Mercurial, the Spectre, Abundant numbers coprime to n, Hi.gher. Space.
• Wikipedia, Dirichlet's theorem on arithmetic progressions

Crossrefs

Cf. A053228, A053223, A000203 (sigma), A358396.

Cf. also A358412, A358413.

Programs

PARI

isA358395(n) = (n%2) && (sigma(n) + sigma(n+2) > 2*sigma(n+1))