A159939 Odd solutions of phi(sigma(k)) = sigma(phi(k)).

1, 9, 225, 729, 18225, 65025, 140625, 531441, 5267025, 11390625, 13286025, 18792225, 40640625, 87890625, 1522170225, 2197265625, 3291890625, 3839661225, 5430953025, 7119140625, 8303765625, 11745140625, 25400390625

Offset: 1

Author

Walter Nissen, Apr 26 2009

Keywords

• nonn

Comments

sigma is the multiplicative sum-of-divisors function.

phi is Euler's totient.

Complete through 25558816403.

All given here are products of powers of consecutive Fermat primes based on generalized repunit primes; see links.

It is conjectured (see links) that all odd solutions are of this form, for which at least 10130 solutions are known.

a(24) > 10^11, if it exists. - Amiram Eldar, Nov 21 2024

Examples

			sigma(9) = 13, phi(9) = 6, sigma(6) = phi(13) = 12, so 9 is in the sequence.
		

References

• Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B42, pp. 150-152.
• Oystein Ore, Number Theory and Its History, 1948, reprinted 1988, Dover, ISBN-10: 0486656209, pp. 88 et seq., 109 et seq.

Links

• Walter Nissen, phi(sigma(n)) = sigma(phi(n)).

Crossrefs

Cf. A000203, A000010, A033632, A019434.

Programs

PARI

isok(n) = (n % 2) && (eulerphi(sigma(n)) == sigma(eulerphi(n))) \\ Michel Marcus, Jul 23 2013

Extensions

Edited by Charles R Greathouse IV, Oct 28 2009

a(1) = 1 inserted by Amiram Eldar, Nov 21 2024