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A005278 Noncototients: n such that x - phi(x) = n has no solution.
(Formerly M4688)
20
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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, 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

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

REFERENCES

J. Browkin and A. Schinzel, On integers not of the form n-phi(n), Colloq. Math., 68 (1995), 55-58.

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

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

LINKS

T. D. Noe and Donovan Johnson, Table of n, a(n) for n = 1..10000 (first 963 terms from T. D. Noe)

Gerry Myerson, Western Number Theory Problems, 17 & 19 Dec 2012

C. Pomerance and H.-S. Yang, On untouchable numbers and related problems, 2012

C. Pomerance and H.-S. Yang, Variant of a theorem of Erdos on the sum-of-proper-divisors function, 2012

Eric Weisstein's World of Mathematics, Noncototient

MATHEMATICA

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 *)

CROSSREFS

Cf. A006093, A126887. Complement of A051953.

Sequence in context: A043342 A023715 A045143 * A157075 A245021 A045039

Adjacent sequences:  A005275 A005276 A005277 * A005279 A005280 A005281

KEYWORD

nonn,nice

AUTHOR

N. J. A. Sloane.

EXTENSIONS

More terms from Jud McCranie 1/97.

STATUS

approved

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Last modified July 24 06:42 EDT 2014. Contains 244876 sequences.