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

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

Author

N. J. A. Sloane

Keywords

• nonn
• nice

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

Examples

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

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.

Crossrefs

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

Programs

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

Magma

[n: n in [1..1000] | EulerPhi(n+2) eq EulerPhi(n)+2]; // Vincenzo Librandi, Sep 11 2015

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 *)

PARI

isok(n) = eulerphi(n+2) == eulerphi(n) + 2; \\ Michel Marcus, Sep 11 2015

Extensions

More terms from Jud McCranie, Dec 24 1999