

A055467


Nonprime numbers for which phi(n) + sigma(n) is an integer multiple of the cube of the number of divisors.


1



1, 95, 99, 121, 125, 159, 287, 319, 415, 447, 511, 543, 654, 671, 703, 767, 799, 831, 895, 959, 1055, 1119, 1247, 1343, 1390, 1495, 1535, 1631, 1727, 1849, 1919, 1983, 2043, 2047, 2060, 2261, 2271, 2335, 2463, 2495, 2559, 2623, 2815, 2828, 2883, 2911
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OFFSET

1,2


COMMENTS

Makowski proved that phi(n) + sigma(n) = nd(n) if and only if n is a prime (see in Sivaramakrishnan, Chapter I, page 8, Theorem 3). In more special cases, k differs from n and phi(n) + sigma(n) is divisible by higher powers of the number of divisors.


REFERENCES

Sivaramakrishnan, R. (1989): Classical Theory of Arithmetical Functions, Marcel Dekker, Inc., New YorkBasel.


LINKS

Matthew House, Table of n, a(n) for n = 1..10000


FORMULA

Integer solutions of phi(x) + sigma(x) = k*d(x)^3 or A000203(n) + A000010(n) = k*A000005(n)^3, where k is an integer.


EXAMPLE

n = 95 with 4 divisors, sigma(95) = 120, phi(95) = 72 72 + 120 = 192 = 3 * 4 * 4 * 4, k = 3.


MATHEMATICA

Select[Range[10000], ! PrimeQ[#] && Mod[EulerPhi[#] + DivisorSigma[1, #], DivisorSigma[0, #]^3] == 0 &] (* Matthew House, Dec 28 2016 *)


CROSSREFS

Cf. A000005, A000010, A000203.
Sequence in context: A033415 A067266 A171403 * A057654 A181767 A046005
Adjacent sequences: A055464 A055465 A055466 * A055468 A055469 A055470


KEYWORD

nonn


AUTHOR

Labos Elemer, Jun 27 2000


EXTENSIONS

Definition corrected by Matthew House, Dec 28 2016


STATUS

approved



