A333039 Composites m such that sigma(m) < sigma(m-1).

9, 21, 25, 27, 33, 35, 39, 45, 46, 49, 51, 55, 57, 65, 69, 77, 81, 85, 87, 91, 93, 95, 99, 105, 106, 111, 115, 117, 118, 119, 121, 123, 125, 129, 133, 141, 143, 145, 153, 155, 159, 161, 165, 166, 169, 171, 175, 177, 183, 185, 187, 189, 201

Offset: 1

Author

Bernard Schott, Mar 12 2020

Keywords

• nonn

Comments

As all primes p >= 5 satisfy sigma(p) < sigma(p-1) [see A333038], this sequence is reserved for composite numbers.

This sequence is infinite because all squares of primes p, p >= 3 are terms.

Composites such that sigma(m) = sigma(m-1) are in A231546.

Examples

			sigma(77) = 1+7+11+77 = 96 < sigma(76) = 1+2+4+19+38+76 = 140, hence composite 77 is a term.
sigma(135) = 1+3+5+9+15+27+45+135 = 240 > sigma(134) = 1+2+67+134 = 204, hence composite 135 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

Cf. A000203, A053222, A231546, A333038.

Programs

Maple

filter:= m -> not isprime(m) and numtheory:-sigma(m) < numtheory:-sigma(m-1) : select(filter, [$1..500]);

Mathematica

Select[Range[200], CompositeQ[#] && DivisorSigma[1, #] < DivisorSigma[1, # - 1] &] (* Amiram Eldar, Mar 12 2020 *)

PARI

isok(m) = (m>1) && !isprime(m) && (sigma(m) < sigma(m-1)); \\ Michel Marcus, Mar 15 2020