A104906 -id:A104906 - cp's OEIS frontend

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

Offset: 1

Farideh Firoozbakht, Apr 13 2005

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

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

Cf. A063903, A104904, A104906.

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

Keywords full, fini from Max Alekseyev, Feb 08 2010

A104904 Numbers n such that d(n)*pi(n)=n, where d(n) is the number of positive divisors of n.

Original entry on oeis.org

2, 8408, 481044, 189961452, 75370122528, 75370124832, 4086199302976, 221945984411264

Offset: 1

Farideh Firoozbakht, Apr 12 2005

Next term is greater than 3*10^9.

			189961452 is in the sequence because d(189961452)=18; pi(189961452)=10553414 & 18*10553414=189961452.

Cf. A063903, A104905, A104906.

Do[If[DivisorSigma[0, n]*PrimePi[n] == n, Print[n]], {n, 2000000000}]

a(5)-a(8) from Donovan Johnson, Dec 08 2009

cp's OEIS Frontend

A104905 Numbers m such that d(m)*phi(m) = sigma(m), where d(m) is number of positive divisors of m.

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