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A001494 Numbers k such that phi(k) = phi(k+2).
(Formerly M3293 N1328)
17
4, 7, 8, 10, 26, 32, 70, 74, 122, 146, 308, 314, 386, 512, 554, 572, 626, 635, 728, 794, 842, 910, 914, 1015, 1082, 1226, 1322, 1330, 1346, 1466, 1514, 1608, 1754, 1994, 2132, 2170, 2186, 2306, 2402, 2426, 2474, 2590, 2642, 2695, 2762, 2906, 3242, 3314 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
If p and 2p-1 are odd primes then 2*(2p-1) is a solution of the equation. Other terms (7,8,32,70,...) are not of this form.
There are 506764111 terms under 10^12. - Jud McCranie, Feb 13 2012
If 2^(2^m) + 1 is a Fermat prime in A019434, so, m = 0, 1, 2, 3, 4, then k = 2^(2^m + 1) is a term; this subsequence consists of {4, 8, 32, 512, 131072} and, in this case, phi(k) = phi(k+2) = 2^(2^m). - Bernard Schott, Apr 22 2022
REFERENCES
D. M. Burton, Elementary Number Theory, section 7-2.
R. K. Guy, Unsolved Problems Number Theory, Sect. B36.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Jud McCranie, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
Kevin Ford, Solutions of phi(n)=phi(n+k) and sigma(n)=sigma(n+k), arXiv:2002.12155 [math.NT], 2020.
M. F. Hasler, Table of n, a(n) for n = 1..17286. (Terms up to 10^7.)
V. L. Klee, Jr., Some remarks on Euler's totient function, Amer. Math. Monthly, 54 (1947), 332.
Leo Moser, Some equations involving Euler's totient function, Amer. Math. Monthly, 56 (1949), 22-23.
FORMULA
A000010(a(n)) = A000010(a(n) + 2). - Reinhard Zumkeller, Feb 08 2013
MATHEMATICA
Select[Range[3500], EulerPhi[#]==EulerPhi[#+2]&] (* Harvey P. Dale, Apr 24 2011 *)
Flatten[Position[Partition[EulerPhi[Range[3500]], 3, 1], _?(#[[1]]==#[[3]]&), {1}, Heads->False]] (* This program is more efficient than the first program above because it only has to calculate phi of each number once. *) (* Harvey P. Dale, Aug 20 2014 *)
SequencePosition[EulerPhi[Range[4300]], {x_, _, x_}][[All, 1]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Sep 04 2020 *)
PROG
(PARI) op=[0, c=0]; for( n=1, 1e7, if( op[bittest(n, 0)+1]+0==op[bittest(n, 0)+1]=eulerphi(n), write("b001494.txt", c++, " "n-2))) \\ M. F. Hasler, Jan 05 2011
(Haskell)
import Data.List (elemIndices)
a001494 n = a001494_list !! (n-1)
a001494_list = map (+ 1) $ elemIndices 0 $
zipWith (-) (drop 2 a000010_list) a000010_list
-- Reinhard Zumkeller, Feb 08 2013
(Magma) [n: n in [1..4000] | EulerPhi(n) eq EulerPhi(n+2)]; // Vincenzo Librandi, Sep 07 2016
CROSSREFS
Sequence in context: A084791 A310933 A186712 * A092214 A319926 A128373
KEYWORD
nonn,nice
AUTHOR
EXTENSIONS
More terms from Jud McCranie, Dec 24 1999
STATUS
approved

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