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A005278
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Noncototients: numbers k such that x - phi(x) = k has no solution.
(Formerly M4688)
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29
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10, 26, 34, 50, 52, 58, 86, 100, 116, 122, 130, 134, 146, 154, 170, 172, 186, 202, 206, 218, 222, 232, 244, 260, 266, 268, 274, 290, 292, 298, 310, 326, 340, 344, 346, 362, 366, 372, 386, 394, 404, 412, 436, 466, 470, 474, 482, 490, 518, 520
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OFFSET
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1,1
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COMMENTS
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Browkin & Schinzel show that this sequence is infinite. - Labos Elemer, Dec 21 1999
If the strong Goldbach conjecture (every even number > 6 is the sum of at least 2 distinct primes p and q) is true, the sequence contains only even values, since p*q - phi(p*q) = p+q-1 and then every odd number can be expressed as x-phi(x). - Benoit Cloitre, Mar 03 2002
Browkin & Schinzel and Hee-sung Yang (Myerson link, problem 012.17d) ask if this sequence has a positive lower density. - Charles R Greathouse IV, Nov 04 2013
Sierpiński (1959) asked if this sequence is infinite.
Erdős (1973) asked if this sequence has a positive lower density.
Browkin and Schinzel (1995) proved that 509203*2^k is a term for all k>=1.
Flammenkamp and Luca (2000) proved that 509203 can be replaced with any other term of A263958 (and found 6 more terms of A263958).
Banks and Luca (2004) proved that the relative density of primes p within the sequence of primes such that 2*p is noncototient is 1. (End)
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REFERENCES
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Richard K. Guy, Unsolved Problems in Number Theory, 3rd ed., Springer, 2004, section B36, pp. 138-142.
Wacław Sierpiński, Number Theory, Part II, PWN Warszawa, 1959 (in Polish).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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FORMULA
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MATHEMATICA
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nmax = 520; cototientQ[n_?EvenQ] := (x = n; While[test = x - EulerPhi[x] == n ; Not[test || x > 2*nmax], x++]; test); cototientQ[n_?OddQ] = True; Select[Range[nmax], !cototientQ[#]&] (* Jean-François Alcover, Jul 20 2011 *)
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PROG
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(PARI) lista(nn)=v = vecsort(vector(nn^2, n, n - eulerphi(n)), , 8); for (n=1, nn, if (! vecsearch(v, n), print1(n, ", "))); \\ Michel Marcus, Oct 03 2016
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CROSSREFS
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Cf. A063740 (number of k such that cototient(k) = n).
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KEYWORD
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nonn,nice
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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