A104905 - OEIS

A104905

Numbers m such that d(m)*phi(m) = sigma(m), where d(m) is number of positive divisors of m.

1, 3, 14, 42

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

d(m)*phi(m) is the product of f(p^k) = (p^k - p^(k-1))*(1+k), while sigma(m) is the product of g(p^k) = (p^(k+1)-1)/(p-1) taken over all prime powers p^k in the factorization of m. We have f(p^k) < g(p^k) for p=2 and k=1 or 2; f(p^k) = g(p^k) for p=3, k=1; and f(p^k) > g(p^k) in all other cases. Furthermore, f(2)/g(2) = 2/3 and f(2^2)/g(2^2) = 6/7, while f(p^k)/g(p^k) > f(p)/g(p) and for p > 7, f(p)/g(p) > 3/2. It easily follows that 1,3,14,42 are the only terms of this sequence. - Max Alekseyev, Feb 08 2010

LINKS

Table of n, a(n) for n=1..4.

EXAMPLE

42 is in the sequence because d(42)=8; phi(42)=12; sigma(42)=96 & 8*12=96.

MATHEMATICA

Do[If[DivisorSigma[0, n]*EulerPhi[n] == DivisorSigma[1, n], Print[n]], {n, 530000000}]

CROSSREFS

Cf. A063903, A104904, A104906.

Sequence in context: A196236 A213482 A296267 * A055650 A367985 A000550

Adjacent sequences: A104902 A104903 A104904 * A104906 A104907 A104908

KEYWORD

nonn,full,fini

AUTHOR

Farideh Firoozbakht, Apr 13 2005

EXTENSIONS

Keywords full, fini from Max Alekseyev, Feb 08 2010

STATUS

approved