A001274 Numbers k such that phi(k) = phi(k+1). (Formerly M2999 N1215)

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

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.

Jonathan Bayless and Paul Kinlaw, Consecutive coincidences of Euler’s function, International Journal of Number Theory, Vol. 12, No. 4 (2016), pp. 1011-1026.

Farideh Firoozbakht, Puzzle 466. phi(n-1)=phi(n)=phi(n+1), in C. Rivera's Primepuzzles.

Kevin Ford, Solutions of phi(n)=phi(n+k) and sigma(n)=sigma(n+k), arXiv:2002.12155 [math.NT], 2020.

Dov Jarden, Recurring Sequences, Riveon Lematematika, Jerusalem, 1966. [Annotated scanned copy] See p. 66.

Paul Kinlaw, Mitsuo Kobayashi and Carl Pomerance, On the equation phi(n) = phi(n+1), Acta Arithmetica, Vol. 196 (2020), pp. 69-92, alternative link.

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

Sum_{n>=1} 1/a(n) is in the interval (1.4324884, 7.8358) (Kinlaw et al., 2020; an upper bound 441702 was given by Bayless and Kinlaw, 2016). - Amiram Eldar, Oct 15 2020

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[200000]], 2, 1], {x_, x_}]] (* Harvey P. Dale, Dec 27 2015 *)
Select[Range[1000], EulerPhi[#] == EulerPhi[# + 1] &] (* Alonso del Arte, Oct 03 2014 *)
SequencePosition[EulerPhi[Range[200000]], {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
(PARI) list(lim)=my(v=List(), old=1); forfactored(n=2, lim\1+1, my(new=eulerphi(n)); if(old==new, listput(v, n[1]-1)); old=new); Vec(v) \\ Charles R Greathouse IV, Jul 17 2022
(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: A343884 A358738 A331689 * A139766 A003276 A136092

Adjacent sequences: A001271 A001272 A001273 * A001275 A001276 A001277

Keyword

nonn,easy,nice

Author

N. J. A. Sloane

Extensions

More terms from David W. Wilson

Status

approved