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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.
0
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
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
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