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%I #25 Mar 25 2024 06:45:42
%S 1,2,4,15,39,49,55,78,81,87,95,99,110,111,119,121,125,143,159,183,184,
%T 215,247,287,295,303,319,327,335,350,357,391,407,415,423,430,447,455,
%U 471,507,511,519,527,535,543,551,559,583,591,620,623,654,655,671,679
%N Numbers k such that d(k)^2 divides phi(k) + sigma(k).
%C 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).
%C 2 is the only prime term of this sequence.
%C Contains p^2 if p is a prime == 2 or 7 (mod 9), and p*q if p and q are distinct primes with p*q == 7 (mod 8). - _Robert Israel_, Jan 18 2018
%D R. Sivaramakrishnan, Classical Theory of Arithmetic Functions, Marcel Dekker, Inc., New York and Basel, 1989.
%H Robert Israel, <a href="/A055466/b055466.txt">Table of n, a(n) for n = 1..10000</a>
%H A. Makowski, <a href="https://gdz.sub.uni-goettingen.de/id/PPN378850199_0013?tify=%7B%22pages%22%3A%5B119%5D%2C%22view%22%3A%22info%22%7D">Problem 339</a>, Elemente der Mathematik, Vol. 13 (1958), p. 115; <a href="https://www.e-periodica.ch/digbib/view?pid=edm-001%3A1958%3A13%3A%3A121">alternative link</a>.
%H C. A. Nicol, <a href="http://www.jstor.org/stable/2312203">Problem E 1674</a>, The American Mathematical Monthly, Vol. 71, No. 3 (1964), p. 317; <a href="http://www.jstor.org/stable/2310997">Another characterization of prime number</a>, Solutions to Problem E 1674 by Martin J. Cohen and J. A. Fridy, ibid., Vol. 72, No. 2 (1965), pp. 186-187.
%F Integer solutions of phi(x) + sigma(x) = m * d(x) or A000010(x) + A000203(x) = m * A000005(x)^2, where m is an integer.
%e 78 is a term since it has 8 divisors, phi(78) = 24, sigma(78) = 168, and 24 + 168 = 192 = 3 * 8^2.
%p filter:= proc(n) uses numtheory;
%p phi(n)+sigma(n) mod tau(n)^2 = 0
%p end proc:
%p select(filter, [$1..1000]); # _Robert Israel_, Jan 18 2018
%t okQ[n_] := Divisible[EulerPhi[n] + DivisorSigma[1, n], DivisorSigma[0, n]^2];
%t Select[Range[1000], okQ] (* _Jean-François Alcover_, May 11 2023 *)
%o (PARI) is(n) = {my(f = factor(n)); (eulerphi(f) + sigma(f)) % numdiv(f)^2 == 0; } \\ _Amiram Eldar_, Mar 25 2024
%Y Cf. A000005, A000010, A000203, A055465, A055467, A055468.
%K nonn
%O 1,2
%A _Labos Elemer_, Jun 27 2000
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