A333040 - OEIS

%I #36 Aug 31 2021 04:37:20

%S 46,106,118,166,226,274,298,316,346,358,406,466,514,526,562,586,622,

%T 694,706,766,778,826,838,862,886,946,1006,1114,1126,1156,1186,1198,

%U 1282,1306,1366,1396,1426,1486,1522,1546,1576,1594,1618,1702,1726,1756

%N Even numbers m such that sigma(m) < sigma(m-1).

%C The even terms of A333039 represent about only 7% of the data, so they are proposed in this sequence. Some of these integers are semiprimes with for example these two families:

%C 1) m = 2*p with p prime of the form k^2+k+3 is in A027753. The first few terms are: 46, 118, 226, 766, ... but not all the integers of this form are terms; the first 3 counterexamples are 6, 10, 1018 (see examples).

%C 2) m = 2*p with p prime = (r*s*t+1)/2 and 2<r<s<t primes, is in A234103. The first few terms are: 106, 166, 274, 346, 358, ... but not all the integers of this form are terms; the first 3 counterexamples are 386, 898 and 958 (see examples).

%C There is also this subsequence of even m = 2^2*p where p prime, congruent to 34 mod 45, is in A142330. The first few terms are: 316, 1396, 1756, 2416, ... but not all the integers of this form are terms; the first counterexample that comes from the 8th term of A142330 is 5716.

%C Even (and odd) numbers such that sigma(m) = sigma(m-1) are in A231546.

%D J.-M. De Koninck & A. Mercier, 1001 Problèmes en Théorie Classique des Nombres, Problème 620 pp. 82, 280, Ellipses Paris 2004.

%H Robert Israel, <a href="/A333040/b333040.txt">Table of n, a(n) for n = 1..10000</a>

%e 166 = 2*83 and 165 = 3*5*11, as 252 = sigma(166) < sigma(165) = 288, hence 166 is a term.

%e 386 = 2*193 and 385 = 5*7*11, as 582 = sigma(386) > sigma(385)= 576, hence 386 is not a term.

%e 766 = 2*383 where 383 = 19^2+19+3 and 765 = 3^2*5*13, as 1152 = sigma(766) < sigma(765) = 1404, hence 766 is a term.

%e 1018 = 2*509 where 509 = 22^2+22+3, and 1017 = 3^2*113, as 1530 = sigma(1018) > sigma(1017) = 1482, hence 1018 is not a term.

%p filter:= n -> numtheory:-sigma(n) < numtheory:-sigma(n-1):

%p select(filter, [seq(i,i=2..2000,2)]); # _Robert Israel_, Mar 29 2020

%t Select[2 * Range[1000], DivisorSigma[1, #] < DivisorSigma[1, #-1] &] (* _Amiram Eldar_, Mar 24 2020 *)

%o (PARI) isok(m) = !(m%2) && (sigma(m) < sigma(m-1)); \\ _Michel Marcus_, Mar 22 2020

%Y Intersection of A005843 and A333039.

%Y Subsequence of A333038.

%Y Cf. A231546.

%K nonn

%O 1,1

%A _Bernard Schott_, Mar 22 2020