OFFSET
1,1
COMMENTS
Sándor (2005) proved that this sequence is infinite by showing that it includes all the numbers of the form 11 * p^11 * k where p != 11 is a prime and k is any number coprime to 11*p.
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..10000
József Sándor, Selected Chapters of Geometry, Analysis and Number Theory, 2005, pp. 132-134.
EXAMPLE
84 is a term since d(84) = 12, sigma(84) = 224 and phi(84) = 24 are all abundant numbers: sigma(12) = 28 > 2*12 = 24, sigma(224) = 504 > 2*224 = 448 and sigma(24) = 60 > 2*24 = 48.
MATHEMATICA
abQ[n_] := DivisorSigma[1, n] > 2*n; q[n_] := And @@ abQ /@ Join[DivisorSigma[{0, 1}, n], {EulerPhi[n]}]; Select[Range[500], q]
PROG
(PARI) isab(k) = sigma(k) > 2*k; \\ A005101
isok(k) = my(f=factor(k)); isab(numdiv(f)) && isab(sigma(f)) && isab(eulerphi(f)); \\ Michel Marcus, Dec 03 2021
CROSSREFS
KEYWORD
nonn
AUTHOR
Amiram Eldar, Nov 29 2021
STATUS
approved