A001494 Numbers k such that phi(k) = phi(k+2). (Formerly M3293 N1328)

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

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

T. D. Noe and 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

Maple

with(numtheory): P:=proc(n) if phi(n)=phi(n+2) then n;
fi; end: seq(P(i), i=1..3400); # Paolo P. Lava, Mar 02 2018

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

Cf. A000010, A001274, A007015, A179186, A179187, A179188, A179189, A179202, A217139.

Sequence in context: A084791 A310933 A186712 * A092214 A319926 A128373

Adjacent sequences: A001491 A001492 A001493 * A001495 A001496 A001497

Keyword

nonn,nice

Author

N. J. A. Sloane

Extensions

More terms from Jud McCranie, Dec 24 1999

Status

approved