A333039 - OEIS

%I #20 Mar 19 2020 13:31:03

%S 9,21,25,27,33,35,39,45,46,49,51,55,57,65,69,77,81,85,87,91,93,95,99,

%T 105,106,111,115,117,118,119,121,123,125,129,133,141,143,145,153,155,

%U 159,161,165,166,169,171,175,177,183,185,187,189,201

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

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

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

%C Composites 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="/A333039/b333039.txt">Table of n, a(n) for n = 1..10000</a>

%e sigma(77) = 1+7+11+77 = 96 < sigma(76) = 1+2+4+19+38+76 = 140, hence composite 77 is a term.

%e 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.

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

%t Select[Range[200], CompositeQ[#] && DivisorSigma[1, #] < DivisorSigma[1, # - 1] &] (* _Amiram Eldar_, Mar 12 2020 *)

%o (PARI) isok(m) = (m>1) && !isprime(m) && (sigma(m) < sigma(m-1)); \\ _Michel Marcus_, Mar 15 2020

%Y Cf. A000203, A053222, A231546, A333038.

%K nonn

%O 1,1

%A _Bernard Schott_, Mar 12 2020