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A033632
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Numbers n such that sigma(phi(n)) = phi(sigma(n)).
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36
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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
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OFFSET
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1,2
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COMMENTS
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The largest term of this sequence that I found is 3^9550. Also, if (1/2)*(3^(n+1)-1) is prime (n+1 is a term of A028491) then 3^n is in the sequence, namely sigma(phi(3^n) = phi(sigma(3^n)) (the proof is easy). - Farideh Firoozbakht, Feb 09 2005
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REFERENCES
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R. K. Guy, Unsolved Problems in Number Theory, 2nd edition, Springer Verlag, 1994, section B42, p.99.
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LINKS
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T. D. Noe and 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S. W. Golomb, Equality among number-theoretic functions, Unpublished manuscript.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
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FORMULA
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A062401(a(n)) = A062402(a(n)). - Reinhard Zumkeller, Jan 04 2013
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MAPLE
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with(numtheory); P:=proc(q) local n;
for n from 1 to q do if sigma(phi(n))=phi(sigma(n)) then print(n);
fi; od; end: P(10^6); # Paolo P. Lava, Aug 08 2013
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MATHEMATICA
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Select[ Range[ 10^6 ], DivisorSigma[ 1, EulerPhi[ # ] ] == EulerPhi[ DivisorSigma[ 1, # ] ] & ]
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PROG
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(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
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CROSSREFS
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Cf. A000203, A000010, A028491, A078148.
Sequence in context: A221439 A205568 A264848 * A110260 A036896 A120319
Adjacent sequences: A033629 A033630 A033631 * A033633 A033634 A033635
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KEYWORD
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nonn,nice
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AUTHOR
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Jud McCranie
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STATUS
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approved
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