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

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

Author

Michel Lagneau, Feb 03 2010

Keywords

• nonn

Examples

			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.
		

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..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

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)))^.125) = (phi(phi(n)) + sigma(sigma(n)))^.125 then print (n);fi ; od;

PARI

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

Extensions

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