|
|
A333039
|
|
Composites m such that sigma(m) < sigma(m-1).
|
|
3
|
|
|
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
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,1
|
|
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.
|
|
REFERENCES
|
J.-M. De Koninck & A. Mercier, 1001 Problèmes en Théorie Classique des Nombres, Problème 620 pp. 82, 280, Ellipses Paris 2004
|
|
LINKS
|
|
|
EXAMPLE
|
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.
|
|
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 *)
|
|
PROG
|
(PARI) isok(m) = (m>1) && !isprime(m) && (sigma(m) < sigma(m-1)); \\ Michel Marcus, Mar 15 2020
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|