A001229 Numbers k such that phi(sigma(k)) = k.

1, 2, 8, 12, 128, 240, 720, 6912, 32768, 142560, 712800, 1140480, 1190400, 3345408, 3571200, 5702400, 14859936, 29719872, 50319360, 118879488, 2147483648, 3889036800, 4389396480, 21946982400, 47416320000, 92177326080, 133145026560, 331914240000, 460886630400, 665725132800

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn

Comments

For n=0,1,2,3,4 & 5 2^(2^n-1) is in the sequence because 2^2^n+1 is prime for n=0,1,2,3 & 4 (Fermat primes). - Farideh Firoozbakht, Oct 08 2004

2987228160000, 11681629470720, and 47996928000000 are also in the sequence. - Jud McCranie, Sep 14 2024

References

• J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 128, p. 44, Ellipses, Paris 2008.
• J.-M. De Koninck & A. Mercier, 1001 Problemes en Theorie Classique Des Nombres, Problem 702 pp. 92; 300-1, Ellipses Paris 2004.
• R. K. Guy, Unsolved Problems in Number Theory, B42.

Links

• Leon Alaoglu and Paul Erdős, A conjecture in elementary number theory, Bull. Amer. Math. Soc. 50 (1944), pp. 881-882.
• Graeme L. Cohen, On a conjecture of Makowski and Schinzel, Colloquium Mathematicae, Vol. 74, No. 1 (1997), pp. 1-8. See Notes p. 7.
• Fred W. Helenius, 664 solutions [Broken Link]
• Fred W. Helenius, 664 solutions [From the Wayback machine]
• T. Negadi, The genetic code invariance: when Euler and Fibonacci meet, arXiv preprint arXiv:1406.6092 [q-bio.OT], 2014; Symmetry: Culture and Science, Vol. 25, No. 3, 261-278, 2014.
• Eric Weisstein's World of Mathematics, Totient Function.

Crossrefs

Cf. A000010, A018784, A135240 A373435, A373453, A373454.

Programs

Mathematica

Select[Range[10000], EulerPhi[DivisorSigma[1, #]] == # &] (* T. D. Noe, Jun 26 2012 *)

PARI

is(n)=eulerphi(sigma(n))==n \\ Charles R Greathouse IV, May 15 2013

Formula

phi(A018784), sorted. - David W. Wilson, Oct 18 2012

Extensions

More terms from David W. Wilson, Aug 15 1996 (search was complete only through a(19) = 50319360).

Jud McCranie reports Jun 15 1998 that the terms through a(24) are certain.

a(28) added. Verified sequence is complete through a(28) by Donovan Johnson, Jun 30 2012

More terms from Jud McCranie, Sep 14 2024, Sep 14 2024. Complete through 10^13.