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A172464 Numbers n such that phi(phi(n)) + sigma(sigma(n)) is a 4th power.

%I A172464 #16 Feb 16 2025 08:33:11
%S A172464 9,42,101,339,407,420,471,915,1409,2572,2847,3706,4069,6631,6720,7229,
%T A172464 9212,14051,16641,31453,33067,33146,35701,37425,37675,37911,48016,
%U A172464 48272,53101,55956,56906,68895,73474,75023,83525,84676,86928,94525,101428,101743,115925
%N A172464 Numbers n such that phi(phi(n)) + sigma(sigma(n)) is a 4th power.
%D A172464 W. L. Glaisher, Number-Divisor Tables. British Assoc. Math. Tables, Vol. 8, Camb. Univ. Press, 1940, p. 64.
%D A172464 S. W. Golomb, Equality among number-theoretic functions, Abstract 882-11-16, Abstracts Amer. Math. Soc., 14 (1993), 415-416.
%D A172464 R. K. Guy, Unsolved Problems in Number Theory, B42.
%H A172464 Hiroaki Yamanouchi, <a href="/A172464/b172464.txt">Table of n, a(n) for n = 1..5749</a>
%H A172464 K. Ford, <a href="http://dx.doi.org/10.1090/S1079-6762-98-00043-2">The distribution of totients</a>, Electron. Res. Announc. Amer. Math. Soc. 4 (1998), 27-34.
%H A172464 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/TotientValenceFunction.html">Totient Valence Function</a>
%H A172464 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CarmichaelsTotientFunctionConjecture.html">Carmichael's Totient Function conjecture</a>
%e A172464 phi(phi(9)) + sigma(sigma(9))= 1;
%e A172464 phi(phi(42)) + sigma(sigma(42))= 4^4 = 256;
%e A172464 phi(phi(101)) + sigma(sigma(101))= 4^4 = 256;
%e A172464 phi(phi(339)) + sigma(sigma(339))= 6^4 = 1296.
%p A172464 with(numtheory): for n from 1 to 2000000 do;if floor(( phi(phi(n)) + sigma(sigma(n)))^.25) =( phi(phi(n)) + sigma(sigma(n)))^.25 then print (n);fi ; od;
%t A172464 Select[Range[116000],IntegerQ[Surd[DivisorSigma[1,DivisorSigma[1,#]]+ EulerPhi[ EulerPhi[ #]],4]]&] (* _Harvey P. Dale_, Aug 16 2021 *)
%Y A172464 Cf. A000010, A002180, A032446, A058277.
%K A172464 nonn
%O A172464 1,1
%A A172464 _Michel Lagneau_, Feb 03 2010
%E A172464 a(40)-a(41) from _Hiroaki Yamanouchi_, Sep 19 2014