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%I M2397 N0951 #52 Sep 24 2023 00:04:49
%S 3,5,6,11,12,14,17,18,20,29,41,44,59,62,71,92,101,107,116,137,149,164,
%T 179,191,197,212,227,239,254,269,281,311,332,347,356,419,431,452,461,
%U 521,524,569,599,617,641,659,692,716,764,809,821,827,857,881,932,956
%N Numbers k such that phi(k+2) = phi(k) + 2.
%C 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.
%C Twice Mersenne primes (cf. A000668) are also solutions. - _Vladeta Jovovic_, Feb 14 2002
%D 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 D. M. Burton, Elementary Number Theory, section 7-2.
%D 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).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H T. D. Noe, <a href="/A001838/b001838.txt">Table of n, a(n) for n = 1..10000</a>
%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%H S. W. Graham, J. J. Holt, and C. Pomerance, <a href="https://math.dartmouth.edu/~carlp/phi.pdf">On the solutions to phi(n) = phi(n+k)</a>, Number Theory in Progress, K. Gyory, H. Iwaniec, and J. Urbanowicz, eds., vol. 2, de Gruyter, Berlin and New York, 1999, pp. 867-882.
%H L. Moser, <a href="http://www.jstor.org/stable/2305815">Some equations involving Euler's totient function</a>, Amer. Math. Monthly, 56 (1949), 22-23.
%e phi(18+2) = 8 = phi(18) + 2, so 18 is in the sequence.
%t Select[Range@1000, EulerPhi@(# + 2)== EulerPhi[#] + 2 &] (* _Vincenzo Librandi_, Sep 11 2015 *)
%t Position[Partition[EulerPhi[Range[1000]],3,1],_?(#[[1]]+2 == #[[3]]&), 1, Heads->False]//Flatten (* _Harvey P. Dale_, Oct 04 2017 *)
%o (Haskell)
%o import Data.List (elemIndices)
%o a001838 n = a001838_list !! (n-1)
%o a001838_list = map (+ 1) $ elemIndices 2 $
%o zipWith (-) (drop 2 a000010_list) a000010_list
%o -- _Reinhard Zumkeller_, Feb 21 2012
%o (PARI) isok(n) = eulerphi(n+2) == eulerphi(n) + 2; \\ _Michel Marcus_, Sep 11 2015
%o (Magma) [n: n in [1..1000] | EulerPhi(n+2) eq EulerPhi(n)+2]; // _Vincenzo Librandi_, Sep 11 2015
%Y Cf. A050472, A050473, etc. Essentially the same as A056853.
%K nonn,nice
%O 1,1
%A _N. J. A. Sloane_
%E More terms from _Jud McCranie_, Dec 24 1999
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