A058277 -id:A058277 - cp's OEIS frontend

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

Michel Lagneau, Feb 03 2010

nonn

			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.

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.

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

Cf. A000010, A002180, A032446, A058277.

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;

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

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

A172465 Numbers n such that phi(phi(n)) + sigma(sigma(n)) is an 8th power.

Original entry on oeis.org

42, 101, 6720, 9212, 226570, 276404, 288086, 299668, 339098, 392228, 412276, 423395, 530917, 535759, 559427, 564209, 666181, 2835284, 3592300, 3911744, 4080100, 5980673, 7230960, 8787900, 14960924, 17130550, 23324242, 27449729, 30437729, 33869141, 42073800

Offset: 1

Michel Lagneau, Feb 03 2010

nonn

			phi(phi(9)) + sigma(sigma(9))= 1;
phi(phi(42)) + sigma(sigma(42))= 2^8 = 256;
phi(phi(101)) + sigma(sigma(101))= 2^8 = 256;
phi(phi(6720)) + sigma(sigma(6720))= 4^8 = 65536.

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.

Hiroaki Yamanouchi, Table of n, a(n) for n = 1..267
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
K. Ford, The distribution of totients, Electron. Res. Announc. Amer. Math. Soc. 4 (1998), 27-34.
Eric Weisstein's World of Mathematics, Carmichael's Totient Function conjecture

Cf. A000010, A002180, A032446, A058277.

with(numtheory):for n from 1 to 2000000 do;if floor(( phi(phi(n)) + sigma(sigma(n)))^.125) = (phi(phi(n)) + sigma(sigma(n)))^.125 then print (n);fi ; od;

isok(n) = ispower(eulerphi(eulerphi(n)) + sigma(sigma(n)), 8); \\ Michel Marcus, Sep 20 2014

a(10) corrected and a(18)-a(31) added by Hiroaki Yamanouchi, Sep 19 2014

cp's OEIS Frontend

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

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