A001274 -id:A001274 - cp's OEIS frontend

A342701 a(n) is the second smallest k such that phi(n+k) = phi(k), or 0 if no such solution exists.

3, 7, 5, 14, 9, 34, 7, 16, 15, 26, 11, 68, 39, 28, 15, 32, 33, 72, 25, 40, 35, 56, 17, 101, 45, 37, 45, 56, 29, 152, 31, 61, 39, 56, 35, 144, 37, 61, 39, 74, 41, 128, 35, 88, 45, 161, 47, 192, 49, 82, 51, 74, 95, 216, 43, 97, 75, 203, 59, 304, 91, 88, 63, 122

Offset: 1

Amiram Eldar, Mar 18 2021

nonn

Sierpiński (1956) proved that there is at least one solution for all n>=1.
Schinzel (1958) proved that there are at least two solutions k to phi(n+k) = phi(k) for all n <= 8*10^47. Schinzel and Wakulicz (1959) increased this bound to 2*10^58.
Schinzel (1958) observed that under the prime k-tuple conjecture there is a second solution for all even n.
Holt (2003) proved that there is a second solution for all even n <= 1.38 * 10^26595411.

			a(1) = 3 since the solutions to the equation phi(1+k) = phi(k) are k = 1, 3, 15, 104, 164, ... (A001274), and 3 is the second solution.

Richard K. Guy, Unsolved Problems in Number Theory, 3rd edition, Springer, 2004, section B36, page 138-142.
József Sándor and Borislav Crstici, Handbook of Number theory II, Kluwer Academic Publishers, 2004, Chapter 3, p. 217-219.
Wacław Sierpiński, Sur une propriété de la fonction phi(n), Publ. Math. Debrecen, Vol. 4 (1956), pp. 184-185.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Jeffery J. Holt, The minimal number of solutions to phi(n)=phi(n+k), Math. Comp., Vol. 72, No. 244 (2003), pp. 2059-2061.
Andrzej Schinzel, Sur l'équation phi(x + k) = phi(x), Acta Arith., Vol. 4, No. 3 (1958), pp. 181-184.
Andrzej Schinzel and Andrzej Wakulicz, Sur l'équation phi(x+k)=phi(x). II, Acta Arith., Vol. 5, No. 4 (1959), pp. 425-426.
Eric Weisstein's World of Mathematics, k-Tuple Conjecture.

Cf. A000010, A001259, A001274, A007015, A217199.

f[n_, 0] = 0; f[n_, k0_] := Module[{k = f[n, k0 - 1] + 1}, While[EulerPhi[n + k] != EulerPhi[k], k++]; k]; Array[f[#, 2] &, 100]

a(n) = my(k=1, nb=0); while ((nb += (eulerphi(n+k)==eulerphi(k))) != 2, k++); k; \\ Michel Marcus, Mar 19 2021

A349308 Numbers k such that A321167(k) = A321167(k+1) > 1.

Original entry on oeis.org

80, 135, 296, 343, 375, 624, 728, 1160, 1431, 1592, 1624, 2240, 2295, 2456, 2511, 2624, 2727, 2888, 3429, 3591, 3624, 3752, 3992, 4023, 4184, 4671, 4887, 4913, 5048, 5144, 5264, 5319, 5480, 5696, 6183, 6344, 6375, 6591, 6615, 6776, 6858, 6859, 7479, 7624, 7640

Offset: 1

Amiram Eldar, Nov 14 2021

nonn

Without the restriction that A321167(k) > 1, all the terms of A340152 would be in this sequence.

In contrast to A001274, which has only one known pair of consecutive terms (5186 and 5187), this sequence seems to have many pairs of consecutive terms. The smaller members of these pairs are 6858, 13375, 22625, ...

			80 is a term since A321167(80) = A321167(81) = 3.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A321167, A340152.
Subsequence of A068140.
Similar sequences: A001274, A287055, A293184, A326403, A349307.

f[p_, e_] := p^e - 1; uphi[1] = 1; uphi[n_] := Times @@ f @@@ FactorInteger[n]; fe[p_, e_] := uphi[e]; euphi[n_] := Times @@ fe @@@ FactorInteger[n]; Select[Range[8000], euphi[#] == euphi[# + 1] > 1 &]

A001836 Numbers k such that phi(2k-1) < phi(2k), where phi is Euler's totient function A000010.

Original entry on oeis.org

53, 83, 158, 263, 293, 368, 578, 683, 743, 788, 878, 893, 908, 998, 1073, 1103, 1208, 1238, 1268, 1403, 1418, 1502, 1523, 1658, 1733, 1838, 1943, 1964, 2048, 2063, 2153, 2228, 2243, 2258, 2363, 2393, 2423, 2468, 2558, 2573, 2633, 2657, 2678

Offset: 1

N. J. A. Sloane

Jeffrey Shallit, personal communication.
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).

T. D. Noe, Table of n, a(n) for n = 1..1000
V. L. Klee, Jr., Some remarks on Euler's totient function, Amer. Math. Monthly, 54 (1947), 332.
Don Reble, Python program.
J. Shallit, Letter to N. J. A. Sloane, Jul 17 1975.

Cf. A000010, A001837.

a001836 n = a001836_list !! (n-1)
a001836_list = f a000010_list 1 where
   f (u:v:ws) x = if u < v then x : f ws (x + 1) else f ws (x + 1)
-- Reinhard Zumkeller, Jul 11 2014

with(numtheory): A001836:=n->`if`(phi(2*n-1) < phi(2*n), n, NULL): seq(A001836(n), n=1..5*10^3); # Wesley Ivan Hurt, Oct 10 2014

Select[Range[3000], EulerPhi[2# - 1] < EulerPhi[2#] &] (* Harvey P. Dale, Apr 01 2012 *)
Position[Partition[EulerPhi[Range[6000]],2],?(#[[1]]<#[[2]]&),1,Heads-> False]//Flatten (* _Harvey P. Dale, Jul 02 2021 *)

is(n)=eulerphi(2*n-1)Charles R Greathouse IV, Feb 21 2013

from sympy import totient
def ok(n): return totient(2*n - 1) < totient(2*n) # Indranil Ghosh, Apr 29 2017

Corrected and extended by Don Reble, Jan 04 2007

A057919 Numbers k such that phi(k) divides phi(k+1), where phi(k) is the Euler totient function A000010.

Original entry on oeis.org

1, 2, 3, 4, 6, 12, 15, 16, 18, 36, 72, 90, 96, 104, 108, 154, 162, 164, 192, 194, 255, 256, 286, 364, 432, 486, 495, 576, 584, 702, 768, 792, 804, 924, 975, 1066, 1152, 1260, 1296, 1458, 2146, 2204, 2592, 2625, 2834, 2916, 3255, 3382, 3456, 3705, 3888

Offset: 1

Leroy Quet, Nov 11 2000

nonn

The intersection of this sequence and A057920 is A001274. - Michel Marcus, Sep 14 2015

			6 is included because phi(6) = 2 divides phi(7) = 6.

Amiram Eldar, Table of n, a(n) for n = 1..6255 (terms below 10^10)

Cf. A001274, A057920.

Select[Range[4000], Divisible[EulerPhi[# + 1], EulerPhi[#]] &] (* Amiram Eldar, Jul 13 2019 *)

lista(nn) = for (n=1, nn, if (eulerphi(n+1) % eulerphi(n) == 0, print1(n, ", "))); \\ Michel Marcus, Sep 14 2015

Offset set to 1 by Michel Marcus, Sep 14 2015

A057920 Numbers k such that phi(k+1) divides phi(k), where phi is A000010.

Original entry on oeis.org

1, 3, 5, 13, 15, 35, 37, 61, 73, 104, 119, 157, 164, 193, 194, 255, 277, 313, 397, 421, 455, 457, 495, 527, 541, 545, 584, 613, 629, 661, 665, 673, 733, 757, 877, 975, 997, 1085, 1093, 1153, 1201, 1213, 1237, 1295, 1321, 1381, 1453, 1469, 1621, 1657, 1753

Offset: 1

Leroy Quet, Nov 11 2000

nonn

The intersection of this sequence and A057919 is A001274. - Michel Marcus, Sep 14 2015

			13 is included because phi(14) = 6 divides phi(13) = 12.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A001274, A057919.

for n from 1 to 2000 do
    if modp(numtheory[phi](n),numtheory[phi](n+1)) =0 then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, Sep 14 2015

Select[Range[1800], Divisible[EulerPhi[#], EulerPhi[# + 1]] &] (* Amiram Eldar, Jul 13 2019 *)

lista(nn) = for (n=1, nn, if (eulerphi(n) % eulerphi(n+1) == 0, print1(n, ", "))); \\ Michel Marcus, Sep 14 2015

A201253 Numbers k such that phi(k+1) = 5*phi(k).

Original entry on oeis.org

11242770, 18673200, 77805000, 117138840, 122649450, 278023200, 393513120, 881879460, 2177410830, 2364390210, 3440848320, 3919303080, 5151045900, 5836032510, 7284273360, 8029787220, 8505803460, 12998545560, 13081794180, 13759304790, 14031484740, 14104654410

Offset: 1

Ray Chandler, Nov 28 2011

nonn

Giovanni Resta, Table of n, a(n) for n = 1..92 (terms < 10^12)

Cf. A000010, A001274, A050472, A067143, A172314.

isok(k) = eulerphi(k+1) == 5*eulerphi(k); \\ Michel Marcus, Aug 10 2025

a(9)-a(22) from Donovan Johnson, Nov 29 2011

A257550 Numbers n such that phi(n) = 5*phi(n+1).

Original entry on oeis.org

17907119, 18828809, 31692569, 73421039, 179467469, 322757819, 337567229, 627702389, 975314339, 2537636009, 2722271369, 3328653509, 3917646809, 5529412349, 6369847469, 11179199849, 11201693579, 11363832479, 13442120999, 16781760449, 19751331599, 20002320029

Offset: 1

Ray Chandler, Apr 29 2015

nonn

			phi(17907119) = 16588800 = 5*phi(17907120).

Giovanni Resta, Table of n, a(n) for n = 1..103 (terms < 10^12)

Cf. A001274, A171262, A256907, A256937.

a1={};nmax=10^9;last=EulerPhi[1];n=2;
While[nRay Chandler, Apr 30 2015 *)

a(10)-a(22) from Giovanni Resta, May 11 2015

A291043 Numbers n such that psi(n) = psi(n+1), where psi(n) is Dedekind psi function (A001615).

Original entry on oeis.org

4, 8, 14, 15, 32, 44, 45, 62, 63, 75, 135, 188, 195, 567, 608, 663, 704, 825, 956, 957, 1023, 1034, 1275, 1334, 1484, 1634, 1845, 1935, 2223, 2534, 2685, 2751, 2871, 3195, 3404, 3843, 3915, 4994, 7004, 7315, 7544, 8024, 8055, 9207, 10695, 11205, 11984, 12032

Offset: 1

Amiram Eldar, Aug 16 2017

nonn

The only solutions to psi(n) = psi(n+1) = psi(n+2) below 10^8 are 14, 44, 62, 956.

In this sequence, smallest terms k such that k and k + 1 are both product of m + 1 distinct primes are 14, 1334, 84134, 3571905, 424152105 for 1 <= m <= 5. - Altug Alkan, Aug 17 2017

			4 is in the sequence since psi(4) = psi(5) = 6.

Amiram Eldar, Table of n, a(n) for n = 1..5000

Cf. A001274, A001615.

psi[n_] := If[n < 1, 0, n Sum[MoebiusMu[d]^2 / d, {d, Divisors @ n}]];
Select[Range[12000], psi[#] == psi[# + 1] &]
SequencePosition[Table[If[n<1,0,n Sum[MoebiusMu[d]^2/d,{d,Divisors[n]}]],{n,13000}],{x_,x_}][[All,1]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Feb 22 2018 *)

a001615(n) = n*sumdivmult(n, d, issquarefree(d)/d);
isok(n) = a001615(n)==a001615(n+1) \\ Altug Alkan, Aug 17 2017, after Charles R Greathouse IV at A001615

A299535 Solutions to A000010(x) + A000010(x-1) = A000010(2*x).

Original entry on oeis.org

2, 4, 16, 256, 496, 976, 2626, 3256, 3706, 5188, 11716, 13366, 18316, 22936, 25546, 46216, 49216, 49336, 57646, 65536, 164176, 184636, 198316, 215776, 286996, 307396, 319276, 388246, 397486, 415276, 491536, 568816, 589408, 686986, 840256, 914176, 952576, 983776

Offset: 1

Benoit Cloitre, Feb 27 2018

nonn

All terms are even. Does lim_{n->oo} log(a(n))/log(n) exist?
Are all terms except 2 congruent to 4 (mod 6)? - Robert Israel, Feb 27 2018 [a(3710) = 3044760173456 is the next term after 2 that is congruent to 2 (mod 6). - Amiram Eldar, Jul 17 2022]
Includes the Fermat numbers 2^(2^j)+1 for j = 0..5, but no other terms of A019434. - Benoit Cloitre, corrected by Robert Israel, Mar 02 2018
Even numbers k for which A000010(k) = A000010(k-1). - Robert Israel, Mar 02 2018

Amiram Eldar, Table of n, a(n) for n = 1..5416 (calculated using the b-file at A001274; terms 1..100 from Robert Israel)
Benoit Cloitre, Plot of log(a(n))/log(n+1) for n=1 up to 62

Cf. A000010.

[n: n in [2..10^6] | EulerPhi(n)+EulerPhi(n-1) eq EulerPhi(2*n)]; // Bruno Berselli, Feb 27 2018

select(t -> numtheory:-phi(t)+ numtheory:-phi(t-1)=numtheory:-phi(2*t), [seq(i,i=2..10^6,2)]); # Robert Israel, Feb 27 2018

for(n=2, 200000, if(eulerphi(n) + eulerphi(n-1) == eulerphi(2*n), print1(n, ",")))

More terms from Robert Israel, Feb 27 2018

A300285 The number of solutions to phi(x) = phi(x+1) below 10^n, where phi(x) is the Euler totient function.

Original entry on oeis.org

2, 3, 10, 17, 36, 68, 142, 306, 651, 1267, 2567, 5236, 10755

Offset: 1

Amiram Eldar, Mar 01 2018

Data extracted from A001274.
The terms were calculated by:
a(1)-a(2) - R. Ratat (1917).
a(3) - Victor L. Klee, Jr. (1947).
a(4)-a(5) - Mohan Lal and Paul Gillard (1972).
a(6) - David Ballew, Janell Case and Robert N. Higgins (1975).
a(7)-a(8) - Robert Baillie (1976).
a(9)-a(10) - Sidney West Graham, Jeffrey J. Holt, and Carl Pomerance (1999).
a(11) - T. D. Noe (2009).
a(12) - Jud McCranie (2012).
a(13) - Giovanni Resta (2014).

			Below 10^2 there are 3 solutions x = 1, 3, 15, hence a(2) = 3.

R. Ratat, L'Intermédiaire des Mathématiciens, Vol. 24, pp. 101-102, 1917.

R. Baillie, Table of phi(n) = phi(n+1), Math. Comp., 30 (1976), pp. 189-190.
David Ballew, Janell Case, and Robert N. Higgins, Table of phi(n)= phi(n+1), Math. Comput., Vol. 29, pp. 329-330, 1975.
Sidney West Graham, Jeffrey J. Holt, and Carl Pomerance,On the solutions to phi(n)= phi(n+ k), Number Theory in Progress, Proceedings of the International Conference in Honor of the 60th Birthday of A. Schinzel, Poland, 1997, Walter de Gruyter, 1999, pp. 867-882.
V. L. Klee, Jr., Some remarks on Euler's totient function, Amer. Math. Monthly, 54 (1947), p. 332.
Mohan Lal and Paul Gillard, On the equation phi(n) = phi(n+k), Math. Comp., 26 (1972), pp. 579-583.
Leo Moser, Some equations involving Euler's totient function, Amer. Math. Monthly, 56 (1949), pp. 22-23.

Cf. A000010, A001274.

With[{s = Array[EulerPhi, 10^6]}, Array[Count[Range[10^# - 1], ?(s[[#]] == s[[# + 1]] &)] &, IntegerLength@ Length@ s - 1]] (* _Michael De Vlieger, Mar 04 2018 *)

According to Thomas Ordowski's conjecture in A001274, a(n) ~ 10^(C*n/3), where C = 9/Pi^2 = 0.911891... Numerically it seems that C ~ 0.93.

cp's OEIS Frontend

A342701 a(n) is the second smallest k such that phi(n+k) = phi(k), or 0 if no such solution exists.

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A349308 Numbers k such that A321167(k) = A321167(k+1) > 1.

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A001836 Numbers k such that phi(2k-1) < phi(2k), where phi is Euler's totient function A000010.

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A057919 Numbers k such that phi(k) divides phi(k+1), where phi(k) is the Euler totient function A000010.

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A057920 Numbers k such that phi(k+1) divides phi(k), where phi is A000010.

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A201253 Numbers k such that phi(k+1) = 5*phi(k).

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A257550 Numbers n such that phi(n) = 5*phi(n+1).

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