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A051488
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Numbers k such that phi(k) < phi(k - phi(k)).
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5
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30, 60, 66, 120, 132, 138, 174, 210, 240, 246, 264, 276, 318, 330, 348, 420, 480, 492, 498, 510, 528, 534, 552, 630, 636, 660, 678, 690, 696, 786, 840, 870, 910, 960, 984, 996, 1020, 1038, 1056, 1068, 1074, 1104, 1122, 1146, 1260, 1272, 1320, 1330, 1356
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OFFSET
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1,1
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COMMENTS
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If p is a Sophie Germain prime greater than 3 and n is a natural number then 2^n*3*p is in the sequence. That is because if m = 2^n*3*p then phi(m) = 2^n*(p-1) and phi(m - phi(m)) = phi(2^n*3*p - 2^n*(p-1)) = phi(2^n*(2p+1)) = 2^n*p so phi(m) < phi(m-phi(m)) and m is in the sequence. - Farideh Firoozbakht, Jun 19 2005
Erdős (1980) proposed the problem to prove that this sequence is infinite and has an asymptotic density 0. Grytczuk et al. (2001) proved that this sequence is infinite with an upper asymptotic density < 0.45637. - Amiram Eldar, May 22 2021
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REFERENCES
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Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B42, p. 150.
József Sándor, Dragoslav S. Mitrinovic and Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, Chapter III, p. 209.
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LINKS
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Paul Erdős, Problem P. 294, Canad. Math. Bull., Vol. 23, No. 4 (1980), p. 505.
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MATHEMATICA
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Select[Range[1360], EulerPhi[ # ] < EulerPhi[ # - EulerPhi[ # ]] &] (* Farideh Firoozbakht, Jun 19 2005 *)
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PROG
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(Haskell)
a051488 n = a051488_list !! (n-1)
a051488_list = [x | x <- [2..], let t = a000010 x, t < a000010 (x - t)]
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CROSSREFS
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KEYWORD
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nonn,nice,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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