A003275 -id:A003275 - cp's OEIS frontend

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

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

N. J. A. Sloane

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

			phi(3) = phi(4) = 2, so 3 is in the sequence.
phi(15) = phi(16) = 8, so 15 is in the sequence.

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).

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.

Cf. A000010, A001494, A051953, A003276, A003275, A007015, A179186, A179187, A179188, A179189, A179202, A217139.

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

[n: n in [1..3*10^5] | EulerPhi(n) eq EulerPhi(n+1)]; // Vincenzo Librandi, Apr 14 2015

select(n -> numtheory:-phi(n) = numtheory:-phi(n+1), [$1..10^5]); # Robert Israel, Mar 31 2015

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 *)

is(n)=eulerphi(n)==eulerphi(n+1) \\ Charles R Greathouse IV, Feb 27 2014

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

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

More terms from David W. Wilson

A003276 Numbers k such that the multiplicative group of residues prime to k, M_k, is isomorphic to M_{k+1}.

Original entry on oeis.org

1, 3, 15, 104, 495, 975, 22935, 32864, 57584, 131144, 491535, 2539004, 3988424, 6235215, 7378371, 13258575, 17949434, 25637744, 26879684, 29357475, 32235735, 41246864, 48615735, 184611375, 229944855, 257278724, 290849624, 429461864, 550666515, 671054835, 706075095

Offset: 1

N. J. A. Sloane

K. Miller, Solutions of phi(n) = phi(n+1) for 1 <= n <= 500000. Unpublished, 1972. [Cf. Math. Comp., Vol. 27, p. 447, 1973.]
D. Shanks, Solved and Unsolved Problems in Number Theory, 2nd. ed., Chelsea, 1978, p. 225.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

P. J. Cameron and D. A. Preece, Notes on primitive lambda-roots.
K. Miller, The equation phi(n) = phi(n+1), Unpublished M.S., ND.
K. Miller, Solutions of phi(n) = phi(n+1) for 1 <= n <= 500000, Mathematics of Computation 27 (1973), 47-48. (Annotated scanned copy)

Subsequence of A001274.

{my(z=znstar(1));for(n=1,10^10,my(z1=znstar(n+1)); if(z[1]==z1[1]&&z[2]==z1[2],print1(n,", "));z=z1;); } \\ Joerg Arndt, Mar 17 2016

list(lim)=my(v=List(),old=[1,[]]); forfactored(n=2,lim\1+1, my(cur=znstar(n)[1..2]); if(old==cur, listput(v,n[1]-1)); old=cur); Vec(v) \\ Charles R Greathouse IV, Jul 17 2022

More terms from David W. Wilson

A290304 Values of uphi(k) = uphi(k+1).

Original entry on oeis.org

1, 12, 24, 120, 96, 180, 432, 744, 720, 864, 840, 1200, 1260, 1680, 2520, 1728, 2784, 2880, 3744, 4032, 5040, 2592, 4224, 5040, 5760, 11520, 11880, 9216, 18000, 20160, 17280, 12480, 17280, 20160, 28080, 20160, 23040, 21600, 32256, 30240, 52080, 34560, 57600

Offset: 1

Amiram Eldar, Jul 26 2017

nonn

The values of unitary totient function of numbers such that k and k+1 have the same value.

The unitary version of A003275.

Amiram Eldar, Table of n, a(n) for n = 1..2198

Cf. A001274, A003275, A047994, A287055.

uphi[n_] := If[n==1, 1, (Times @@ (Table[#[[1]]^#[[2]] - 1, {1}] & /@ FactorInteger[n]))[[1]]]; a={}; u1=0; For[k=0, k<10^5, k++; u2=uphi[k]; If[u1==u2, a = AppendTo[a, u1]]; u1=u2]; a

a(n) = A047994(A287055(n)).

A294030 Values of bphi(k) = bphi(k+1), where bphi is the bi-unitary analog of Euler's totient function (A116550).

Original entry on oeis.org

1, 9, 14, 42, 161, 161, 798, 1400, 86156, 123656, 419430, 387868, 508797, 772121, 870233, 4162866, 8754569, 126168912, 126991491, 128007618, 131491736

Offset: 1

Amiram Eldar, Oct 22 2017

The bi-unitary totient function of numbers k such that k and k+1 have the same function value (A293184).

			9 is in the sequence since 9 = bphi(14) = bphi(15).

The bi-unitary version of A003275.

Cf. A116550, A293184.

bphi[1] = 1; bphi[n_] := With[{pp = Power @@@ FactorInteger[n]}, Count[Range[n], m_ /; Intersection[pp, Power @@@ FactorInteger[m]] == {}]]; a={}; b1=0; Do[b2 = bphi[k]; If[b1 == b2, a = AppendTo[a, b1]]; b1 = b2, {k, 1, 10^2}]; a (* after Jean-François Alcover at A116550 *)

a(n) = A116550(A293184(n)).

a(10)-a(11) from Michel Marcus, Nov 14 2017

a(12)-a(21) from Amiram Eldar, Jul 16 2022

cp's OEIS Frontend

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

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A003276 Numbers k such that the multiplicative group of residues prime to k, M_k, is isomorphic to M_{k+1}.

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A290304 Values of uphi(k) = uphi(k+1).

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Programs

Mathematica

Formula

A294030 Values of bphi(k) = bphi(k+1), where bphi is the bi-unitary analog of Euler's totient function (A116550).

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Keywords

Comments

Examples

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Programs

Mathematica

Formula

Extensions