OFFSET
1,2
COMMENTS
Sándor (2005) proved that this sequence is infinite by showing that any number of the form 2^(p-1), where p is a sufficiently large prime, is a term. d(2^(p-1)) = p and phi(2^(p-1)) = 2^(p-2) are deficient for all primes, while sigma(2^(p-1)) = 2^p - 1 is deficient for a sufficiently large prime, a result of a theorem by Bojanić (1954): lim_{p prime -> oo} sigma(2^p - 1)/(2^p - 1) = 1.
REFERENCES
R. Bojanić, Asymptotic evaluations of the sum of divisors of certain numbers (in Serbo-Croatian), Bull. Soc. Math.-Phys. R. P. Macédoine, Vol. 5 (1954), pp. 5-15.
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
2 is a term since d(2) = 2, sigma(2) = 3 and phi(2) = 1 are all deficient numbers.
MATHEMATICA
defQ[n_] := DivisorSigma[1, n] < 2*n; q[n_] := And @@ defQ /@ Join[DivisorSigma[{0, 1}, n], {EulerPhi[n]}]; Select[Range[10^5], q]
PROG
(PARI) isdef(k) = sigma(k) < 2*k;
isok(k) = my(f=factor(k)); isdef(numdiv(f)) && isdef(sigma(f)) && isdef(eulerphi(k)); \\ Michel Marcus, Dec 01 2021
CROSSREFS
KEYWORD
nonn
AUTHOR
Amiram Eldar, Nov 29 2021
STATUS
approved