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A033632
Numbers k such that sigma(phi(k)) = phi(sigma(k)).
38
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
1,2
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S. W. Golomb, Equality among number-theoretic functions, Unpublished manuscript.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
FORMULA
A062401(a(n)) = A062402(a(n)). - Reinhard Zumkeller, Jan 04 2013
MATHEMATICA
Select[ Range[ 10^6 ], DivisorSigma[ 1, EulerPhi[ # ] ] == EulerPhi[ DivisorSigma[ 1, # ] ] & ]
PROG
(Haskell)
a033632 n = a033632_list !! (n-1)
a033632_list = filter (\x -> a062401 x == a062402 x) [1..]
-- Reinhard Zumkeller, Jan 04 2013
(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
CROSSREFS
KEYWORD
nonn,nice
AUTHOR
STATUS
approved