A055470 Smallest number x > 1 such that phi(x) + sigma(x) = k*d(x)^n, i.e., the left-hand side is divisible by the n-th power of the number of divisors.

2, 2, 95, 121, 121, 2047, 49151, 98303, 393215, 1572863, 6291455, 8388607, 201326591, 805306367, 3221225471, 6442450943, 137438953471, 137438953471, 137438953471

Offset

1,1

Comments

It appears that for n > 5, a(n) is a semiprime. - Lambert Klasen (Lambert.Klasen(AT)gmx.net), Aug 22 2005

a(20) <= 2199023255551. a(21) <= 2199023255551. - Donovan Johnson, Jun 05 2011

Links

Table of n, a(n) for n=1..19.

Formula

Least integer x > 1 such that A000010(x) + A000203(x) = k*A000005(x)^n.

Example

The terms of list {2,2,95,121,121,2047,49151,98303} have {2,2,4,3,3,4,4,4} divisors, {3,3,120,133,133,2160,51312,99000} divisor-sums, {1,1,72,110,110,1936,46992,97608} EulerPhi values. The phi+sigma sums are {4,4,192,243,243,4096,98304,196608}, which are divisible by {2,4,64,81,243,4096,16384,65536} increasing powers of d-numbers, giving {2,1,3,3,1,1,6,3} quotients respectively.

Mathematica

Join[{2, 2}, Table[n = 3; While[! Divisible[(DivisorSigma[1, n] + EulerPhi[n]), DivisorSigma[0, n]^i], n += 2]; n, {i, 3, 12}]] (* Jayanta Basu, Jul 12 2013 *)

Prog

(PARI) k=2; for(n=1, 15, while(denominator((sigma(k)+eulerphi(k))/(sigma(k, 0)^n))!=1, k++); \ print(n, " ", k)) (Klasen)

Crossrefs

Cf. A000005, A000010, A000203.

Sequence in context: A156511 A166996 A133295 * A270591 A210467 A156524

Adjacent sequences: A055467 A055468 A055469 * A055471 A055472 A055473

Keyword

nonn

Author

Labos Elemer, Jun 27 2000

Extensions

More terms from Jud McCranie, Oct 08 2000

More terms from Lambert Klasen (Lambert.Klasen(AT)gmx.net), Aug 22 2005

a(16)-a(19) from Donovan Johnson, Jun 05 2011

Status

approved