A217139 -id:A217139 - 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

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

Original entry on oeis.org

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

N. J. A. Sloane

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

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).

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.

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

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

[n: n in [1..4000] | EulerPhi(n) eq EulerPhi(n+2)]; // Vincenzo Librandi, Sep 07 2016

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 *)

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

A000010(a(n)) = A000010(a(n) + 2). - Reinhard Zumkeller, Feb 08 2013

More terms from Jud McCranie, Dec 24 1999

A179186 Numbers k such that phi(k) = phi(k+4), with Euler's totient function phi = A000010.

Original entry on oeis.org

8, 14, 16, 20, 35, 52, 64, 91, 140, 148, 244, 292, 403, 455, 616, 628, 772, 801, 1011, 1024, 1108, 1144, 1252, 1270, 1295, 1456, 1588, 1684, 1820, 1828, 2030, 2164, 2452, 2623, 2644, 2660, 2692, 2932, 3028, 3216, 3321, 3508, 3988, 4264, 4340, 4372, 4612, 4804, 4852, 4948

Offset: 1

M. F. Hasler, Jan 05 2011

nonn

Is there some k > 5 such that phi(k) = phi(k+3)?
None up to 500000. - Harvey P. Dale, Feb 16 2011
No further solutions to the phi(k) = phi(k+3) problem less than 10^12. On the other hand, this sequence has 267797240 terms under 10^12. - Jud McCranie, Feb 13 2012
No reason is known that would prevent other solutions of phi(k) = phi(k+3), see Graham, Holt, & Pomerance. - Jud McCranie, Jan 03 2013
If a(n) is even then a(n)/2 is in A001494 - see comment at A217139. - Jud McCranie, Dec 31 2012

S. W. Graham, J. J. Holt, and C. Pomerance, "On the solutions to phi(n)=phi(n+k)", Number Theory in Progress, Proc. Intern. Conf. in Honor of 60th Birthday of A. Schinzel, Poland, 1997. Walter de Gruyter, 1999, pp. 867-82.

Jud McCranie, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
F. 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.

Cf. A000010, A001274, A001494, A179187, A007015.

[n: n in [1..5000] | EulerPhi(n) eq EulerPhi(n+4)]; // Vincenzo Librandi, Sep 08 2016

Select[Range[5000],EulerPhi[#]==EulerPhi[#+4]&]  (* Harvey P. Dale, Feb 16 2011 *)
SequencePosition[EulerPhi[Range[5000]],{x_,,,_,x_}][[;;,1]] (* Harvey P. Dale, Sep 12 2024 *)

{op=vector(N=4); for( n=1,1e4,if( op[n%N+1]+0==op[n%N+1]=eulerphi(n),print1(n-N,",")))}

A179202 Numbers n such that phi(n) = phi(n+8), with Euler's totient function phi=A000010.

Original entry on oeis.org

13, 16, 19, 25, 28, 32, 40, 70, 104, 128, 175, 182, 209, 280, 296, 488, 551, 584, 657, 715, 806, 910, 1232, 1256, 1544, 1602, 2022, 2048, 2216, 2288, 2504, 2540, 2590, 2717, 2912, 3176, 3368, 3640, 3656, 4060, 4328, 4904, 5246, 5288, 5320, 5384, 5864, 5969

Offset: 1

M. F. Hasler, Jan 05 2011

nonn

Among the 5596 terms below 10^7, a(6)=32 is the only term such that a(n+1) = a(n)+8.
There are 141741552 terms under 10^12. - Jud McCranie, Feb 13 2012
If a(n) is even then a(n)/2 is in A179186 - see comment at A217139. - Jud McCranie, Dec 31 2012

M. F. Hasler and Jud McCranie, Table of n, a(n) for n = 1..10000 (first 5596 terms from M. F. Hasler)
F. 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.

Cf. A000010, A001274, A001494, A179186, A179187, A179188, A179189, A007015.

[n: n in [1..10000] | EulerPhi(n) eq EulerPhi(n+8)]; // Vincenzo Librandi, Sep 08 2016

Select[Range[6000], EulerPhi[#] == EulerPhi[# + 8] &] (* Vincenzo Librandi, Sep 08 2016 *)

{op=vector(N=8); for( n=1, 1e4, if( op[n%N+1]+0==op[n%N+1]=eulerphi(n), print1(n-N, ", ")))}

A000010(a(n)) = A000010(a(n)+8).

A217141 Numbers n such that phi(n) = phi(n+12) and n is not divisible by 2.

Original entry on oeis.org

143, 157, 203, 247, 273, 1147, 1209, 1679, 2147, 2279, 2375, 2445, 2705, 2747, 4331, 4687, 5049, 6107, 7367, 7835, 7869, 7979, 7991, 9167, 12127, 17145, 18501, 18753, 18981, 19803, 22987, 26733, 27359, 29097, 29987, 32829, 35485, 35763, 37653, 37851, 39907

Offset: 1

Michel Marcus, Sep 27 2012

Donovan Johnson, Table of n, a(n) for n = 1..1000

Cf. A000010, A179188, A217139.

Select[Range[1,40000,2],EulerPhi[#]==EulerPhi[#+12]&] (* Harvey P. Dale, Aug 22 2025 *)

{op=vector(N=12); Nd6=N/6;for( n=1, 1e4, if( op[n%N+1]+0==op[n%N+1]=eulerphi(n), if ((n-N) % Nd6 != 0, print1(n-N, ", "))))}

Definition clarified by Harvey P. Dale, Aug 22 2025

A276503 Numbers n such that phi(n) = phi(n+10), with Euler's totient function phi = A000010.

Original entry on oeis.org

20, 26, 35, 100, 130, 160, 370, 400, 610, 730, 793, 1000, 1570, 1843, 1930, 2500, 2560, 2770, 2860, 3130, 3970, 4000, 4171, 4210, 4570, 5410, 5767, 6130, 6400, 6610, 6730, 7330, 7570, 8770, 9106, 9640, 9970, 9991, 10498, 10660, 10930, 11248

Offset: 1

Vincenzo Librandi, Sep 08 2016

Vincenzo Librandi, Table of n, a(n) for n = 1..1200
Kevin Ford, Solutions of phi(n) = phi(n+k) and sigma(n) = sigma(n + k), arXiv:2002.12155 [math.NT], 2020.

Cf. A000010.

Cf. numbers n such that phi(n)=phi(n+k): A001274 (k=1), A001494 (k=2), A179186 (k=4), A179187 (k=5), A179188 (k=6), A179189 (k=7), A179202 (k=8), this sequence (k=10), A276504 (k=11), A217139 (k=12).

[n: n in [1..20000] | EulerPhi(n) eq EulerPhi(n+10)];

Select[Range[15000], EulerPhi[#] == EulerPhi[# + 10] &]
SequencePosition[EulerPhi[Range[12000]],{x_,,,_,,,_,,,_,x_}][[;;,1]] (* Harvey P. Dale, Apr 29 2025 *)

A217140 a(n) = m/n where m is the least number divisible by n such that phi(m) = phi(m+6n).

Original entry on oeis.org

24, 24, 24, 24, 36, 24, 24, 24, 24, 36, 24, 24, 24, 24, 36, 24, 24, 24, 24, 36, 24, 24, 24, 24, 36, 24, 24, 24, 24, 36, 24, 24, 24, 24, 60, 24, 24, 24, 24, 36, 24, 24, 24, 24, 36, 24, 24, 24, 24, 36, 24, 24, 24, 24, 36, 24, 24, 24, 24, 36, 24, 24, 24, 24, 36

Offset: 1

Michel Marcus and Jonathan Sondow, Oct 01 2012

nonn

			A179188(1)=24 is divisible by 1 and the quotient is 24, so a(1)=24.
A217139(1)=48 is divisible by 2 and the quotient is 24, so a(2)=24.
The first solution to phi(n)=phi(n+18) to be divisible by 3 is 72 and the quotient is 24, so a(3)=24.

Michel Marcus, Table of n, a(n) for n = 1..1000

Cf. A007015, A179188, A217139, A217141, A217068.

A217142 Least number m such that phi(m) = phi(m+6n) and m is not divisible by n.

Original entry on oeis.org

143, 52, 101, 124, 104, 123, 183, 156, 248, 144, 208, 267, 241, 365, 219, 248, 312, 306, 496, 369, 288, 432, 241, 543, 369, 468, 482, 386, 730, 444, 438, 432, 496, 1220, 624, 779, 612, 801, 915, 744, 723, 582, 576, 1095, 864, 488, 482, 641, 1086, 674, 738

Offset: 2

Michel Marcus, Sep 27 2012

nonn

Donovan Johnson, Table of n, a(n) for n = 2..1000

Cf. A000010, A179188, A217068, A217139, A217141.

fpr(Nmax, lim) = {for (i=2, Nmax,N = i*6;op = vector(N);f = 0;for (n=1, lim, if (op[n%N+1]+0==op[n%N+1]=eulerphi(n), if ((n-N) % i != 0, f = n-N;break;);););print1(f, ", "););}

A330251 Numbers k such that phi(k) = phi(k+3), where phi (A000010) is Euler's totient function.

Original entry on oeis.org

3, 5, 8720288051472, 9134280520365, 41544070492925, 42466684755492, 51363581614342, 68616494581632, 113312918293575, 210911076210835, 215517565688425, 294988451482725, 383617980270525, 432759876053505, 442863123838135, 532068058516992, 892813363927485, 923102743748185, 929531173876305

Offset: 1

Michel Marcus and Giovanni Resta, Feb 29 2020

nonn

10^15 < a(20) <= 1089641067389872.
Also terms: 1248817919303952, 1332436545865422, 1394926716616125, 1868522795664525, 1950445682260072.
a(4) and a(9) appear in Kevin Ford's paper.

Kevin Ford, Solutions of phi(n)=phi(n+k) and sigma(n)=sigma(n+k), arXiv:2002.12155 [math.NT], 2020.
S. W. Graham, J. J. Holt, and C. Pomerance, On the solutions to phi(n) = phi(n+k), Number Theory in Progress, K. Gyory, H. Iwaniec, and J. Urbanowicz, eds., vol. 2, de Gruyter, Berlin and New York, 1999, pp. 867-882.
Mathematics StackExchange, Conjecture on the gap between integers having the same number of co-primes, Sep 25 2019.

Cf. A000010, A007015.
Cf. A001274, A001494, A179186, A179187, A179188, A179189, A179202, A330429.
Cf. A276503, A276504, A217139.

Select[Range[100000], EulerPhi[#] == EulerPhi[# + 3] &] (* Alonso del Arte, Mar 01 2020 *)

isok(k) = eulerphi(k) == eulerphi(k+3); \\ Michel Marcus, Feb 29 2020

A330429 Numbers k such that phi(k) = phi(k+9), where phi (A000010) is Euler's totient function.

Original entry on oeis.org

9, 15, 1005079920836, 13695542245376, 26160864154416, 27402841561095, 27599063056565, 110263115897935, 124632211478775, 127400054266476, 154090744843026, 205849483744896, 231019991767556, 339938754880725, 459718637643265, 632733228632505, 646552697065275, 683008674773416, 884965354448175

Offset: 1

Giovanni Resta, Mar 01 2020

nonn

a(20) > 10^15.

Kevin Ford, Solutions of phi(n)=phi(n+k) and sigma(n)=sigma(n+k), arXiv:2002.12155 [math.NT], 2020.
S. W. Graham, J. J. Holt, and C. Pomerance, On the solutions to phi(n) = phi(n+k), Number Theory in Progress, K. Gyory, H. Iwaniec, and J. Urbanowicz, eds., vol. 2, de Gruyter, Berlin and New York, 1999, pp. 867-882.

Cf. A000010, A007015.
Cf. A001274, A001494, A330251, A179186, A179187, A179188, A179189, A179202.
Cf. A276503, A276504, A217139.

cp's OEIS Frontend

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

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

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A179186 Numbers k such that phi(k) = phi(k+4), with Euler's totient function phi = A000010.

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A179202 Numbers n such that phi(n) = phi(n+8), with Euler's totient function phi=A000010.

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A217141 Numbers n such that phi(n) = phi(n+12) and n is not divisible by 2.

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A276503 Numbers n such that phi(n) = phi(n+10), with Euler's totient function phi = A000010.

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Magma

Mathematica

A217140 a(n) = m/n where m is the least number divisible by n such that phi(m) = phi(m+6n).

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