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

9, 42, 101, 339, 407, 420, 471, 915, 1409, 2572, 2847, 3706, 4069, 6631, 6720, 7229, 9212, 14051, 16641, 31453, 33067, 33146, 35701, 37425, 37675, 37911, 48016, 48272, 53101, 55956, 56906, 68895, 73474, 75023, 83525, 84676, 86928, 94525, 101428, 101743, 115925

Offset: 1

Author

Michel Lagneau, Feb 03 2010

Keywords

• nonn

Examples

			phi(phi(9)) + sigma(sigma(9))= 1;
phi(phi(42)) + sigma(sigma(42))= 4^4 = 256;
phi(phi(101)) + sigma(sigma(101))= 4^4 = 256;
phi(phi(339)) + sigma(sigma(339))= 6^4 = 1296.
		

References

• W. L. Glaisher, Number-Divisor Tables. British Assoc. Math. Tables, Vol. 8, Camb. Univ. Press, 1940, p. 64.
• S. W. Golomb, Equality among number-theoretic functions, Abstract 882-11-16, Abstracts Amer. Math. Soc., 14 (1993), 415-416.
• R. K. Guy, Unsolved Problems in Number Theory, B42.

Links

• Hiroaki Yamanouchi, Table of n, a(n) for n = 1..5749
• K. Ford, The distribution of totients, Electron. Res. Announc. Amer. Math. Soc. 4 (1998), 27-34.
• Eric Weisstein's World of Mathematics, Totient Valence Function
• Eric Weisstein's World of Mathematics, Carmichael's Totient Function conjecture

Crossrefs

Cf. A000010, A002180, A032446, A058277.

Programs

Maple

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;

Mathematica

Select[Range[116000],IntegerQ[Surd[DivisorSigma[1,DivisorSigma[1,#]]+ EulerPhi[ EulerPhi[ #]],4]]&] (* Harvey P. Dale, Aug 16 2021 *)

Extensions

a(40)-a(41) from Hiroaki Yamanouchi, Sep 19 2014