A055468 Composite numbers k for which phi(k) + sigma(k) is an integer multiple of the 4th power of the number of divisors of k.

121, 125, 511, 767, 895, 1535, 1919, 2047, 2559, 2815, 3071, 3199, 3327, 3455, 3711, 3839, 4223, 4351, 4479, 4607, 4735, 4863, 5262, 5631, 5726, 5759, 5902, 5966, 6014, 6527, 7167, 7295, 7423, 7679, 7807, 8063, 9599, 9727, 9819, 9983, 10239

Offset

1,1

Comments

Makowski proved that phi(k) + sigma(k) = k*d(k) if and only if k is a prime (see in Sivaramakrishnan, Chapter I, page 8, Theorem 3). Generally, when phi(k) + sigma(k) = m*d(k) there are special cases in which phi(k) + sigma(k) is divisible by higher powers of the number of divisors d(k).

This sequence is infinite: it includes all the semiprimes p*q such that p == 1 (mod 128), and q == 127 (mod 128). - Amiram Eldar, Mar 25 2024

References

R. Sivaramakrishnan, Classical Theory of Arithmetic Functions, Marcel Dekker, Inc., New York and Basel, 1989.

Links

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

A. Makowski, Problem 339, Elemente der Mathematik, Vol. 13 (1958), p. 115; alternative link.

C. A. Nicol, Problem E 1674, The American Mathematical Monthly, Vol. 71, No. 3 (1964), p. 317; Another characterization of prime number, Solutions to Problem E 1674 by Martin J. Cohen and J. A. Fridy, ibid., Vol. 72, No. 2 (1965), pp. 186-187.

Formula

Integer solutions of phi(x) + sigma(x) = m * d(x)^4 or A000010(x) + A000203(x) = m * A000005(k)^4, where m is an integer.

Example

511 is a term since it has 4 divisors, phi(511) = 432, sigma(511) = 592, and 432 + 592 = 1024 = 4 * 4^4.

Mathematica

Select[Range[10000], CompositeQ[#] && Divisible[EulerPhi[#] + DivisorSigma[1, #], DivisorSigma[0, #]^4] &] (* Amiram Eldar, Mar 25 2024 *)

Prog

(PARI) is(n)=my(f=factor(n)); (eulerphi(f)+sigma(f))%numdiv(f)^4==0 && !isprime(n) \\ Charles R Greathouse IV, Mar 01 2017

Crossrefs

Cf. A000005, A000010, A000203, A055465, A055466, A055467.

Sequence in context: A137850 A262517 A036231 * A134328 A113614 A134941

Adjacent sequences: A055465 A055466 A055467 * A055469 A055470 A055471

Keyword

nonn

Author

Labos Elemer, Jun 27 2000

Status

approved