login
This site is supported by donations to The OEIS Foundation.

 

Logo

Annual Appeal: Please make a donation to keep the OEIS running. In 2018 we replaced the server with a faster one, added 20000 new sequences, and reached 7000 citations (often saying "discovered thanks to the OEIS").
Other ways to donate

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A001274 Numbers n such that phi(n) = phi(n+1).
(Formerly M2999 N1215)
41
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

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

T. D. Noe and Jud McCranie, Table of n, a(n) for n = 1..5236 (first 2567 terms from T. D. Noe)

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.

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-Fran├žois 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 *)

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

N. J. A. Sloane

EXTENSIONS

More terms from David W. Wilson

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified December 14 19:37 EST 2018. Contains 318107 sequences. (Running on oeis4.)