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A005278 Noncototients: n such that x - phi(x) = n has no solution.
(Formerly M4688)
23
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

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

REFERENCES

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)

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

A. Flammenkamp and F. Luca, Infinite families of noncototients, Colloq. Math., 86 (2000), 37-41.

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

FORMULA

A005278 = { k | A063740(k) = 0 }. - M. F. Hasler, Jan 11 2018

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 *)

PROG

(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

CROSSREFS

Cf. A006093, A126887. Complement of A051953.

Cf. A063740 (number of k such that cototient(k) = n).

Sequence in context: A043342 A023715 A045143 * A157075 A262998 A245021

Adjacent sequences:  A005275 A005276 A005277 * A005279 A005280 A005281

KEYWORD

nonn,nice

AUTHOR

N. J. A. Sloane.

EXTENSIONS

More terms from Jud McCranie, Jan 01 1997

STATUS

approved

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Last modified December 13 17:36 EST 2018. Contains 318086 sequences. (Running on oeis4.)