A003275 Values of phi(k) when phi(k) = phi(k+1). (Formerly M1874)

1, 2, 8, 48, 80, 96, 128, 240, 288, 480, 1008, 1200, 1296, 1440, 1728, 2592, 2592, 4800, 5600, 6480, 8640, 11040, 12480, 14976, 19008, 19200, 22464, 24320, 24576, 21120, 28416, 27840, 25920, 32000, 32768, 36000, 47520, 52992, 60480, 59904, 79200, 89280, 96768

Offset

1,2

Comments

In other words, consider k = 1,2,3,4,..., and if phi(k) = phi(k+1), add phi(k) to the sequence.

References

Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B36, pp. 138-142.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Amiram Eldar, Table of n, a(n) for n = 1..10755 (calculated from the b-file at A001274; terms 1..2567 from T. D. Noe)

Kathryn Miller, The equation phi(n) = phi(n+1), Unpublished M.S., ND..

Kathryn Miller, Solutions of phi(n) = phi(n+1) for 1 <= n <= 500000. Unpublished, 1972. [ See Review, Math. Comp., Vol. 27, p. 447-448, 1973 ].

Kathryn Miller, Solutions of phi(n) = phi(n+1) for 1 <= n <= 500000, Mathematics of Computation 27 (1973), 47-48. (Annotated scanned copy)

Leo Moser, Some equations involving Euler's totient function, Amer. Math. Monthly, 56 (1949), 22-23.

Eric Weisstein's World of Mathematics, Totient Function.

Formula

a(n) = A000010(A001274(n)). - Reinhard Zumkeller, May 20 2014

Mathematica

Cases[Split[Table[EulerPhi[k], {k, 1, 50000}]], {_, _}][[1;; 27, 1]] (* Jean-François Alcover, Mar 20 2011 *)
#[[1]]&/@Select[Partition[EulerPhi[Range[80000]], 2, 1], #[[1]]==#[[2]]&] (* Harvey P. Dale, Oct 03 2012 *)
SequenceCases[EulerPhi[Range[200000]], {x_, x_}][[All, 1]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Mar 05 2019 *)

Prog

(Haskell)
a003275 = a000010 . fromIntegral . a001274
-- Reinhard Zumkeller, May 20 2014
(PARI) lista(lim) = my(p1 = 1, p2); for(k = 2, lim, p2 = eulerphi(k); if(p1 == p2, print1(p1, ", ")); p1 = p2); \\ Amiram Eldar, Nov 27 2024

Crossrefs

Cf. A000010, A001274.

Sequence in context: A380236 A181413 A358822 * A253665 A078558 A003032

Adjacent sequences: A003272 A003273 A003274 * A003276 A003277 A003278

Keyword

nonn,nice

Author

N. J. A. Sloane

Status

approved