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Numbers n such that phi(phi(n)) + sigma(sigma(n)) is a 4th power.
1

%I #15 Aug 16 2021 17:12:16

%S 9,42,101,339,407,420,471,915,1409,2572,2847,3706,4069,6631,6720,7229,

%T 9212,14051,16641,31453,33067,33146,35701,37425,37675,37911,48016,

%U 48272,53101,55956,56906,68895,73474,75023,83525,84676,86928,94525,101428,101743,115925

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

%D W. L. Glaisher, Number-Divisor Tables. British Assoc. Math. Tables, Vol. 8, Camb. Univ. Press, 1940, p. 64.

%D S. W. Golomb, Equality among number-theoretic functions, Abstract 882-11-16, Abstracts Amer. Math. Soc., 14 (1993), 415-416.

%D R. K. Guy, Unsolved Problems in Number Theory, B42.

%H Hiroaki Yamanouchi, <a href="/A172464/b172464.txt">Table of n, a(n) for n = 1..5749</a>

%H 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 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/TotientValenceFunction.html">Totient Valence Function</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CarmichaelsTotientFunctionConjecture.html">Carmichael's Totient Function conjecture</a>

%e phi(phi(9)) + sigma(sigma(9))= 1;

%e phi(phi(42)) + sigma(sigma(42))= 4^4 = 256;

%e phi(phi(101)) + sigma(sigma(101))= 4^4 = 256;

%e phi(phi(339)) + sigma(sigma(339))= 6^4 = 1296.

%p 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 Select[Range[116000],IntegerQ[Surd[DivisorSigma[1,DivisorSigma[1,#]]+ EulerPhi[ EulerPhi[ #]],4]]&] (* _Harvey P. Dale_, Aug 16 2021 *)

%Y Cf. A000010, A002180, A032446, A058277.

%K nonn

%O 1,1

%A _Michel Lagneau_, Feb 03 2010

%E a(40)-a(41) from _Hiroaki Yamanouchi_, Sep 19 2014