A001838 Numbers k such that phi(k+2) = phi(k) + 2. (Formerly M2397 N0951)

3, 5, 6, 11, 12, 14, 17, 18, 20, 29, 41, 44, 59, 62, 71, 92, 101, 107, 116, 137, 149, 164, 179, 191, 197, 212, 227, 239, 254, 269, 281, 311, 332, 347, 356, 419, 431, 452, 461, 521, 524, 569, 599, 617, 641, 659, 692, 716, 764, 809, 821, 827, 857, 881, 932, 956

Offset

1,1

Comments

If p and p+2 are primes then p is a solution. If p and 2p+1 are both odd primes then 4p is a solution. Several numbers of the form 2^j-2 are solutions (see cross-referenced sequences). Although 18 is a solution, it is not of any of these forms.

Twice Mersenne primes (cf. A000668) are also solutions. - Vladeta Jovovic, Feb 14 2002

References

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 840.

D. M. Burton, Elementary Number Theory, section 7-2.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence as N0951, although there are errors, probably caused by errors in the original source).

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

Links

T. D. Noe, Table of n, a(n) for n = 1..10000

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

S. W. Graham, J. J. Holt, and C. Pomerance, On the solutions to phi(n) = phi(n+k), Number Theory in Progress, K. Gyory, H. Iwaniec, and J. Urbanowicz, eds., vol. 2, de Gruyter, Berlin and New York, 1999, pp. 867-882.

L. Moser, Some equations involving Euler's totient function, Amer. Math. Monthly, 56 (1949), 22-23.

Example

phi(18+2) = 8 = phi(18) + 2, so 18 is in the sequence.

Mathematica

Select[Range@1000, EulerPhi@(# + 2)== EulerPhi[#] + 2 &] (* Vincenzo Librandi, Sep 11 2015 *)
Position[Partition[EulerPhi[Range[1000]], 3, 1], _?(#[[1]]+2 == #[[3]]&), 1, Heads->False]//Flatten (* Harvey P. Dale, Oct 04 2017 *)

Prog

(Haskell)
import Data.List (elemIndices)
a001838 n = a001838_list !! (n-1)
a001838_list = map (+ 1) $ elemIndices 2 $
   zipWith (-) (drop 2 a000010_list) a000010_list
-- Reinhard Zumkeller, Feb 21 2012
(PARI) isok(n) = eulerphi(n+2) == eulerphi(n) + 2; \\ Michel Marcus, Sep 11 2015
(Magma) [n: n in [1..1000] | EulerPhi(n+2) eq EulerPhi(n)+2]; // Vincenzo Librandi, Sep 11 2015

Crossrefs

Cf. A050472, A050473, etc. Essentially the same as A056853.

Sequence in context: A327433 A358866 A167522 * A285785 A080759 A145714

Adjacent sequences: A001835 A001836 A001837 * A001839 A001840 A001841

Keyword

nonn,nice

Author

N. J. A. Sloane

Extensions

More terms from Jud McCranie, Dec 24 1999

Status

approved