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A005278 Noncototients: n such that x - phi(x) = n has no solution.
(Formerly M4688)
22

%I M4688

%S 10,26,34,50,52,58,86,100,116,122,130,134,146,154,170,172,186,202,206,

%T 218,222,232,244,260,266,268,274,290,292,298,310,326,340,344,346,362,

%U 366,372,386,394,404,412,436,466,470,474,482,490,518,520

%N Noncototients: n such that x - phi(x) = n has no solution.

%C Browkin & Schinzel show that this sequence is infinite. - _Labos Elemer_, Dec 21 1999

%C If the strong Goldbach conjecture (every even number>6 is the sum of at least 2 distinct primes p and q) is true, 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

%C Hee-sung Yang (Myerson link, problem 012.17d) asks if this sequence has a positive lower density. - _Charles R Greathouse IV_, Nov 04 2013

%D R. K. Guy, Unsolved Problems in Number Theory, B36.

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H T. D. Noe and Donovan Johnson, <a href="/A005278/b005278.txt">Table of n, a(n) for n = 1..10000</a> (first 963 terms from T. D. Noe)

%H J. Browkin and A. Schinzel, <a href="http://matwbn.icm.edu.pl/ksiazki/cm/cm68/cm6817.pdf">On integers not of the form n-phi(n)</a>, Colloq. Math., 68 (1995), 55-58.

%H A. Flammenkamp and F. Luca, <a href="http://matwbn.icm.edu.pl/ksiazki/cm/cm86/cm8616.pdf">Infinite families of noncototients</a>, Colloq. Math., 86 (2000), 37-41.

%H Gerry Myerson, <a href="http://wcnt.files.wordpress.com/2013/09/wcnt-problems-2012.pdf">Western Number Theory Problems</a>, 17 & 19 Dec 2012

%H C. Pomerance and H.-S. Yang, <a href="http://www.math.dartmouth.edu/~carlp/uupaper3.pdf">On untouchable numbers and related problems</a>, 2012

%H C. Pomerance and H.-S. Yang, <a href="http://www.math.dartmouth.edu/~carlp/uupaper6.pdf">Variant of a theorem of Erdos on the sum-of-proper-divisors function</a>, 2012

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Noncototient.html">Noncototient</a>

%t 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-Fran├žois Alcover_, Jul 20 2011 *)

%o (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

%Y Cf. A006093, A126887. Complement of A051953.

%K nonn,nice

%O 1,1

%A _N. J. A. Sloane_.

%E More terms from _Jud McCranie_, Jan 01 1997

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Last modified December 7 23:55 EST 2016. Contains 278902 sequences.