A033632 Numbers k such that sigma(phi(k)) = phi(sigma(k)).

1, 9, 225, 242, 516, 729, 3872, 13932, 14406, 17672, 18225, 20124, 21780, 29262, 29616, 45996, 65025, 76832, 92778, 95916, 106092, 106308, 114630, 114930, 121872, 125652, 140130, 140625, 145794, 149124, 160986, 179562, 185100, 234876, 248652, 252978, 256860

Offset: 1

Author

Jud McCranie

Keywords

• nonn
• nice

Comments

The largest term of this sequence that I found is 3^9550. Also, if (1/2)*(3^(k+1)-1) is prime (k+1 is a term of A028491) then 3^k is in the sequence, namely sigma(phi(3^k)) = phi(sigma(3^k)) (the proof is easy). - Farideh Firoozbakht, Feb 09 2005

References

• R. K. Guy, Unsolved Problems in Number Theory, 2nd edition, Springer Verlag, 1994, section B42, p. 99.

Links

• Donovan Johnson, Table of n, a(n) for n = 1..10000 (first 200 terms from T. D. Noe)
• S. W. Golomb, Equality among number-theoretic functions, Unpublished manuscript. (Annotated scanned copy)
• Walter Nissen, sigma(phi(n)) = phi(sigma(n)), Up for the Count !
• Walter Nissen, sigma(phi(n)) = phi(sigma(n)): From "5" to "5 figures", Up for the Count !, Nov. 14, 2000
• Walter Nissen, Addendum to : sigma(phi()): From "5" to "5 figures", Up for the Count !, June 8, 2008
• Walter Nissen, Elaboration of : sigma(phi()): From "5" to "5 figures", Up for the Count !, Oct. 15, 2010

Crossrefs

Cf. A000203, A000010, A028491, A078148.

Programs

Haskell

a033632 n = a033632_list !! (n-1)
a033632_list = filter (\x -> a062401 x == a062402 x) [1..]
-- Reinhard Zumkeller, Jan 04 2013

Mathematica

Select[ Range[ 10^6 ], DivisorSigma[ 1, EulerPhi[ # ] ] == EulerPhi[ DivisorSigma[ 1, # ] ] & ]

PARI

is(n)=sigma(eulerphi(n))==eulerphi(sigma(n)) \\ Charles R Greathouse IV, May 09 2013

Python

from sympy import divisor_sigma as sigma, totient as phi
def ok(n): return sigma(phi(n)) == phi(sigma(n))
def aupto(nn): return [m for m in range(1, nn+1) if ok(m)]
print(aupto(10**4)) # Michael S. Branicky, Jan 09 2021

Formula

A062401(a(n)) = A062402(a(n)). - Reinhard Zumkeller, Jan 04 2013