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A104905 Numbers m such that d(m)*phi(m) = sigma(m), where d(m) is number of positive divisors of m. 3
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
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
Sequence in context: A196236 A213482 A296267 * A055650 A367985 A000550
KEYWORD
nonn,full,fini
AUTHOR
Farideh Firoozbakht, Apr 13 2005
EXTENSIONS
Keywords full, fini from Max Alekseyev, Feb 08 2010
STATUS
approved

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Last modified March 29 04:23 EDT 2024. Contains 371264 sequences. (Running on oeis4.)