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%I #35 Mar 25 2024 06:45:46
%S 1,95,99,121,125,159,287,319,415,447,511,543,654,671,703,767,799,831,
%T 895,959,1055,1119,1247,1343,1390,1495,1535,1631,1727,1849,1919,1983,
%U 2043,2047,2060,2261,2271,2335,2463,2495,2559,2623,2815,2828,2883,2911
%N Nonprime numbers k for which phi(k) + sigma(k) is an integer multiple of the cube of the number of divisors of 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). 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).
%C This sequence is infinite: it includes all the semiprimes p*q such that p == 1 (mod 32), and q == 31 (mod 32). - _Amiram Eldar_, Mar 25 2024
%D R. Sivaramakrishnan, Classical Theory of Arithmetic Functions, Marcel Dekker, Inc., New York and Basel, 1989.
%H Matthew House, <a href="/A055467/b055467.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)^3 or A000010(x) + A000203(x) = m * A000005(x)^3, where m is an integer.
%e 95 is a term since it has 4 divisors, phi(95) = 72, sigma(95) = 120, and 72 + 120 = 192 = 3 * 4^3.
%t Select[Range[10000], ! PrimeQ[#] && Mod[EulerPhi[#] + DivisorSigma[1, #], DivisorSigma[0, #]^3] == 0 &] (* _Matthew House_, Dec 28 2016 *)
%o (PARI) is(n) = {my(f = factor(n)); f[,2] != [1]~ && (eulerphi(f) + sigma(f)) % numdiv(f)^3 == 0; } \\ _Amiram Eldar_, Mar 25 2024
%Y Cf. A000005, A000010, A000203, A055465, A055466, A055468.
%K nonn
%O 1,2
%A _Labos Elemer_, Jun 27 2000
%E Definition corrected by _Matthew House_, Dec 28 2016
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