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 A001274 Numbers n such that phi(n) = phi(n+1). (Formerly M2999 N1215) 43
 1, 3, 15, 104, 164, 194, 255, 495, 584, 975, 2204, 2625, 2834, 3255, 3705, 5186, 5187, 10604, 11715, 13365, 18315, 22935, 25545, 32864, 38804, 39524, 46215, 48704, 49215, 49335, 56864, 57584, 57645, 64004, 65535, 73124, 105524, 107864, 123824, 131144, 164175, 184635 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Unlike totients, cototient(x + 1) = cototient(x) never holds (except 2 - phi(2) = 3 - phi(3) = 1) because cototient(x) is congruent to x modulo 2. - Labos Elemer, Aug 08 2001 Lal-Gillard and Firoozbakht ask whether there is another pair of consecutive integers in this sequence, apart from a(16) + 1 = a(17) = 5187, see link. - M. F. Hasler, Jan 05 2011 There are 5236 terms less than 10^12. - Jud McCranie, Feb 13 2012 Up to 10^13 there are 10755 terms, and no further consecutive pairs like (5186, 5187). - Giovanni Resta, Feb 27 2014 A051179(k) for k from 0 to 5 are in the sequence. No other members of A051179 are in the sequence, because phi(2^(2^k)-1) = Product_{j=1..k-1} phi(2^(2^j)+1) and phi(2^(2^5)+1) < 2^(2^5) so if k > 5, phi(2^(2^k)-1) < Product_{j=1..k-1} 2^(2^j) = 2^(2^k-1) = phi(2^(2^k)). - Robert Israel, Mar 31 2015 Number of terms < 10^k, k=1,2,3,...: 2, 3, 10, 17, 36, 68, 142, 306, 651, 1267, 2567, 5236, 10755, ..., . - Robert G. Wilson v, Apr 10 2019 Conjecture: Except for the first two terms, all terms are composite and congruent to either 2 or 3 (mod 6). - Robert G. Wilson v, Apr 10 2019 Paul Kinlaw has noticed that up to 10^13 the only counterexample to the above conjecture is a(7424) = 3044760173455. - Giovanni Resta, May 23 2019 REFERENCES J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 15, pp 5, Ellipses, Paris 2008. 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 Giovanni Resta, Table of n, a(n) for n = 1..10755 (terms < 10^13, a(1)-a(2567) from T. D. Noe, a(2568)-a(5236) from J. McCranie) R. Baillie, Table of phi(n) = phi(n+1), Math. Comp., 30 (1976), 189-190. Farideh Firoozbakht, Puzzle 466. phi(n-1)=phi(n)=phi(n+1), in C. Rivera's Primepuzzles. Dov Jarden, Recurring Sequences, Riveon Lematematika, Jerusalem, 1966. [Annotated scanned copy] See p. 66. Paul Kinlaw, Mitsuo Kobayashi, Carl Pomerance, On the equation phi(n) = phi(n + 1), Husson University (Bangor ME), Dartmouth College (Hanover NH, 2019). V. L. Klee, Jr., Some remarks on Euler's totient function, Amer. Math. Monthly, 54 (1947), 332. M. Lal and P. Gillard, On the equation phi(n) = phi(n+k), Math. Comp., 26 (1972), 579-583. K. Miller, Solutions of phi(n) = phi(n+1) for 1 <= n <= 500000. Unpublished, 1972. [ Cf. (Untitled), Math. Comp., Vol. 27, p. 447, 1973 ]. Leo Moser, Some equations involving Euler's totient function, Amer. Math. Monthly, 56 (1949), 22-23. J. Shallit, Letter to N. J. A. Sloane, Jul 17 1975 FORMULA Conjecture: a(n) ~ C*n^3*log(n), where C = 9/pi^2 = 0.91189... - Thomas Ordowski, Oct 21 2014 EXAMPLE phi(3) = phi(4) = 2, so 3 is in the sequence. phi(15) = phi(16) = 8, so 15 is in the sequence. MAPLE select(n -> numtheory:-phi(n) = numtheory:-phi(n+1), [\$1..10^5]); # Robert Israel, Mar 31 2015 MATHEMATICA Reap[For[n = 1; k = 2; f1 = 1, k <= 10^9, k++, f2 = EulerPhi[k]; If[f1 == f2, Print["a(", n, ") = ", k - 1]; Sow[k - 1]; n++]; f1 = f2]][[2, 1]] (* Jean-François Alcover, Mar 29 2011, revised Dec 26 2013 *) Flatten[Position[Partition[EulerPhi[Range], 2, 1], {x_, x_}]] (* Harvey P. Dale, Dec 27 2015 *) Select[Range, EulerPhi[#] == EulerPhi[# + 1] &] (* Alonso del Arte, Oct 03 2014 *) SequencePosition[EulerPhi[Range], {x_, x_}][[All, 1]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, May 01 2018 *) k = 8; lst = {1, 3}; While[k < 200000, If[ !PrimeQ[k +1], ep = EulerPhi[k +1]; If[ EulerPhi[k] == ep, AppendTo[lst, k]]; If[ep == EulerPhi[k +2], AppendTo[lst, k +1]]]; k += 6]; lst (* Robert G. Wilson v, Apr 10 2019 *) PROG (Haskell) import Data.List (elemIndices) a001274 n = a001274_list !! (n-1) a001274_list = map (+ 1) \$ elemIndices 0 \$                            zipWith (-) (tail a000010_list) a000010_list -- Reinhard Zumkeller, May 20 2014, Mar 31 2011 (PARI) is(n)=eulerphi(n)==eulerphi(n+1) \\ Charles R Greathouse IV, Feb 27 2014 (MAGMA) [n: n in [1..3*10^5] | EulerPhi(n) eq EulerPhi(n+1)]; // Vincenzo Librandi, Apr 14 2015 CROSSREFS Cf. A000010, A001494, A051953, A003276, A003275, A007015, A179186, A179187, A179188, A179189, A179202, A217139. Sequence in context: A123184 A079486 A245118 * A139766 A003276 A136092 Adjacent sequences:  A001271 A001272 A001273 * A001275 A001276 A001277 KEYWORD nonn,easy,nice AUTHOR EXTENSIONS More terms from David W. Wilson STATUS approved

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Last modified October 22 05:29 EDT 2019. Contains 328315 sequences. (Running on oeis4.)