

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
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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*qphi(p*q)=p+q1 and then every odd number can be expressed as xphi(x).  Benoit Cloitre, Mar 03 2002
Browkin & Schinzel and Heesung 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 nphi(n), Colloq. Math., 68 (1995), 5558.
A. Flammenkamp and F. Luca, Infinite families of noncototients, Colloq. Math., 86 (2000), 3741.
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 sumofproperdivisors 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[#]&] (* JeanFrançois 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



