A333040 Even numbers m such that sigma(m) < sigma(m-1).

46, 106, 118, 166, 226, 274, 298, 316, 346, 358, 406, 466, 514, 526, 562, 586, 622, 694, 706, 766, 778, 826, 838, 862, 886, 946, 1006, 1114, 1126, 1156, 1186, 1198, 1282, 1306, 1366, 1396, 1426, 1486, 1522, 1546, 1576, 1594, 1618, 1702, 1726, 1756

Offset: 1

Author

Bernard Schott, Mar 22 2020

Keywords

• nonn

Comments

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:

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).

2) m = 2*p with p prime = (r*s*t+1)/2 and 2A234103. 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).

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.

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

Examples

			166 = 2*83 and 165 = 3*5*11, as 252 = sigma(166) < sigma(165) = 288, hence 166 is a term.
386 = 2*193 and 385 = 5*7*11, as 582 = sigma(386) > sigma(385)= 576, hence 386 is not a term.
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.
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.
		

References

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

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Intersection of A005843 and A333039.

Subsequence of A333038.

Cf. A231546.

Programs

Maple

filter:= n -> numtheory:-sigma(n) < numtheory:-sigma(n-1):
select(filter, [seq(i,i=2..2000,2)]); # Robert Israel, Mar 29 2020

Mathematica

Select[2 * Range[1000], DivisorSigma[1, #] < DivisorSigma[1, #-1] &] (* Amiram Eldar, Mar 24 2020 *)

PARI

isok(m) = !(m%2) && (sigma(m) < sigma(m-1)); \\ Michel Marcus, Mar 22 2020