A001274 -id:A001274 - cp's OEIS frontend

A326403 Numbers k such that iphi(k) = iphi(k+1), where iphi(k) is an infinitary analog to the Euler totient function (A091732).

1, 20, 35, 143, 194, 208, 740, 1220, 1299, 1419, 1803, 1892, 3705, 3716, 3843, 5186, 5635, 7868, 10659, 13634, 13905, 17948, 18507, 18914, 18980, 21007, 25388, 25545, 30380, 31599, 32304, 34595, 37820, 47067, 70394, 73059, 78064, 87856, 94874, 105908, 116963

Offset: 1

Amiram Eldar, Sep 14 2019

nonn

			20 is in the sequence since iphi(20) = iphi(21) = 12.

Amiram Eldar, Table of n, a(n) for n = 1..1000

Cf. A001274, A091732, A287055, A293184, A301866.

f[p_, e_] := p^(2^(-1 + Position[Reverse @ IntegerDigits[e, 2], ?(# == 1 &)])); a[1] = 1; a[n] := Times @@ (Flatten @(f @@@ FactorInteger[n]) - 1); a1 = 1; s = {}; Do[a2 = a[n]; If[a1 == a2, AppendTo[s, n - 1]]; a1 = a2, {n, 2, 10^5}]; s

A057000 a(n) = phi(n+1) - phi(n).

Original entry on oeis.org

0, 1, 0, 2, -2, 4, -2, 2, -2, 6, -6, 8, -6, 2, 0, 8, -10, 12, -10, 4, -2, 12, -14, 12, -8, 6, -6, 16, -20, 22, -14, 4, -4, 8, -12, 24, -18, 6, -8, 24, -28, 30, -22, 4, -2, 24, -30, 26, -22, 12, -8, 28, -34, 22, -16, 12, -8, 30, -42, 44, -30, 6, -4, 16, -28, 46, -34, 12, -20, 46, -46, 48, -36, 4

Offset: 1

N. J. A. Sloane, Sep 09 2000

sign

T. D. Noe, Table of n, a(n) for n = 1..10000
N. J. A. Sloane, Families of Essentially Identical Sequences, Mar 24 2021 (Includes this sequence)

Cf. A000010, A001274.

[(EulerPhi(n+1) - EulerPhi(n)): n in [1..100]]; // Vincenzo Librandi, Sep 30 2013

A057000 := proc(n)
    numtheory[phi](n+1)-numtheory[phi](n) ;
end proc:
seq(A057000(n),n=1..40) ; # R. J. Mathar, May 10 2023

Table[EulerPhi[n + 1] - EulerPhi[n], {n, 100}] (* Vincenzo Librandi, Sep 30 2013 *)

a(n) = eulerphi(n+1) - eulerphi(n); \\ Michel Marcus, Jan 29 2017

G.f.: -1 + (1 - x)*Sum_{k>=1} mu(k)*x^(k-1)/(1 - x^k)^2. - Ilya Gutkovskiy, Jan 29 2017

A001837 Numbers k such that phi(2k+1) < phi(2k).

Original entry on oeis.org

157, 262, 367, 412, 472, 487, 577, 682, 787, 877, 892, 907, 997, 1072, 1207, 1237, 1312, 1402, 1522, 1567, 1627, 1657, 1732, 1852, 1942, 2047, 2062, 2152, 2194, 2257, 2362, 2437, 2467, 2557, 2572, 2677, 2722, 2782

Offset: 1

N. J. A. Sloane

nonn

Greg Martin (gerg(AT)math.toronto.edu) writes: I recently calculated the smallest solution of phi(30k+1) < phi(30k) (see the Martin link); it has 1116 digits.

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 157, p. 51, Ellipses, Paris 2008.
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).

Donovan Johnson, Table of n, a(n) for n = 1..10000
V. L. Klee, Jr., Some remarks on Euler's totient function, Amer. Math. Monthly, 54 (1947), 332.
Greg Martin, The smallest solution of phi(30n+1) < phi(30n) is ..., arXiv:math/9804025 [math.NT], 1998; Amer. Math. Monthly, Vol. 106, No. 5 (1999), pp. 449-451.
J. Shallit, Letter to N. J. A. Sloane, Jul 17 1975

Cf. A000010.

with(numtheory,phi); f := proc(n) if phi(2*n+1) < phi(2*n) then RETURN(n) fi end;

Select[ Range[4000], EulerPhi[2# + 1] < EulerPhi[2# ] & ]

isok(n) = eulerphi(2*n+1) < eulerphi(2*n); \\ Michel Marcus, Oct 03 2017

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

Original entry on oeis.org

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

N. J. A. Sloane

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

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).

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.

Cf. A000010, A001274.

a003275 = a000010 . fromIntegral . a001274
-- Reinhard Zumkeller, May 20 2014

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 *)

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

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

A067143 Numbers n such that phi(n+1) = 3*phi(n).

Original entry on oeis.org

6, 12, 18, 36, 72, 90, 96, 108, 162, 192, 432, 486, 576, 702, 768, 792, 924, 1152, 1296, 1458, 2592, 2916, 3456, 3888, 4698, 5550, 6696, 7998, 8700, 10368, 10590, 11802, 12288, 16470, 17496, 18432, 33250, 39366, 52488, 56790, 79248, 124356

Offset: 1

Benoit Cloitre, Feb 19 2002

nonn

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

Cf. A001274, A050472, A172314.

Position[Partition[EulerPhi[Range[125000]],2,1],?(#[[2]]==3#[[1]]&), 1,Heads-> False]//Flatten (* _Harvey P. Dale, Jul 30 2020 *)

isok(n) = eulerphi(n+1) == 3*eulerphi(n); \\ Michel Marcus, Nov 20 2013

More terms from Dean Hickerson, Feb 20 2002

A172314 Numbers k such that phi(k+1) = 4*phi(k).

Original entry on oeis.org

1260, 13650, 17556, 18720, 24510, 42120, 113610, 244530, 266070, 712080, 749910, 795690, 992250, 1080720, 1286730, 1458270, 1849470, 2271060, 2457690, 3295380, 3370770, 3414840, 3714750, 4061970, 4736490, 5314050, 5827080, 6566910, 6935082, 7303980, 7864080

Offset: 1

Michel Lagneau, Jan 31 2010

nonn

			phi(1260) = 288. phi(1261) = 1152. 4*phi(1260) = phi(1261).

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 51, p. 19, Ellipses, Paris 2008.
R. K. Guy, Unsolved Problems Number Theory, Sect. B36.

Donovan Johnson, Table of n, a(n) for n = 1..300
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. De Pauw University, 1972. [ Cf. Review on Math. Comp., Vol. 27, p. 447, 1973 ].
L. Moser, Some equations involving Euler's totient function, Amer. Math. Monthly, 56 (1949), 22-23.
A. Shinzel, Sur l'équation phi(x+k) = phi(x), Acta Arith. 4 (1958), 181-184, [MR0106857]

Cf. A001274, A050472, A067143.

[n: n in [1..2*10^6] | EulerPhi(n+1) eq 4*EulerPhi(n)]; // Vincenzo Librandi, Jan 27 2016

with(numtheory): for n from 1 to 4000000 do; if 4*phi(n) = phi(n+1) then print(n); else fi ; od;

#[[1,1]]&/@Select[Partition[Table[{n,EulerPhi[n]},{n,4000000}],2,1], 4#[[1,2]]==#[[2,2]]&] (* Harvey P. Dale, Oct 11 2011 *)
Select[Range@1000000, EulerPhi@# 4 == EulerPhi[# + 1] &] (* Vincenzo Librandi, Jan 27 2016 *)

References separated by R. J. Mathar, Feb 19 2010

A266276 a(n) is the smallest number k such that phi(k) = n*phi(k-1).

Original entry on oeis.org

2, 3, 7, 1261, 11242771

Offset: 1

Jaroslav Krizek, Jan 26 2016

a(n) >= A266269(n). - Max Alekseyev, Jan 26 2025

			a(3) = 7 because 7 is the smallest number k such that phi(k) = n*phi(k-1); phi(7) = 6 =3*phi(6) = 3*2.

Sequences of numbers n such that phi(n) = k*phi(n-1): {A001274 + 1} for k=1; A171271 = {A050472 + 1} for k=2; A266268 = {A067143 + 1} for k=3; A268126 = {A172314 + 1} for k=4; {A201253 + 1} for k=5.

Cf. A000010, A091439, A091456, A266269.

Magma
```
a:=func; [a(n):n in[1..5]];
```

a(n) = my(k=2, epk=1, enk); while ((enk=eulerphi(k)) != n*epk, epk = enk; k++); k; \\ Michel Marcus, Feb 20 2020

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 *)

A349307 Numbers k such that A072911(k) = A072911(k+1) > 1.

Original entry on oeis.org

80, 135, 296, 343, 375, 567, 624, 728, 783, 944, 1160, 1431, 1592, 1624, 1647, 1808, 1863, 2024, 2240, 2295, 2456, 2511, 2624, 2727, 2888, 3087, 3320, 3429, 3536, 3591, 3624, 3752, 3992, 4023, 4184, 4239, 4375, 4400, 4455, 4624, 4671, 4887, 4912, 4913, 5048, 5103

Offset: 1

Amiram Eldar, Nov 14 2021

nonn

Without the restriction that A072911(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 4912, 5750, 6858, ...

			80 is a term since A072911(80) = A072911(81) = 2.

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

Cf. A072911, A340152.
Subsequence of A068140.
Similar sequences: A001274, A287055, A293184, A326403, A349308.

f[p_, e_] := EulerPhi[e]; ephi[n_] := Times @@ f @@@ FactorInteger[n]; Select[Range[5000], ephi[#] == ephi[# + 1] > 1 &]

A275412 Least k such that phi(kn) = phi(kn+1), or 0 if no such k exists.

Original entry on oeis.org

1, 52, 1, 26, 3

Offset: 1

Altug Alkan, Jul 27 2016

From Michael De Vlieger, Jul 27 2016: (Start)
Terms 6 through 80, with 0 signifying no such k <= 10^6: {0, 375, 13, 55, 0, 45, 0, 8, 180121, 1, 2054, 15, 0, 116, 0, 125, 482, 2145, 0, 39, 4, 495, 144098, 76, 0, 105, 1027, 15, 37556, 75, 0, 495, 58, 25, 0, 4, 0, 7425, 241, 11, 2294, 178425, 0, 8475, 0, 5, 2, 5415, 0, 9, 72049, 65, 38, 976, 0, 944, 1702, 915, 761, 15, 0, 161175, 18778, 715, 0, 165, 0, 8, 1426, 13, 29, 20451, 0, 416, 0}.
n such that a(n) = 0 in above data: {6, 10, 12, 18, 20, 24, 30, 36, 40, 42, 48, 50, 54, 60, 66, 70, 72, 78, 80}, i.e., multiples of 2 and 3, and 2 and 5. (End)
If a(n) = 0, then a(m*n) = 0 for all m > 0. If c is a divisor of a(n), then a(c*n) = a(n)/c. Conjecture: If A275337(n) = 0, then a(n) = 0. - Chai Wah Wu, Jul 27 2016

			a(2) = 52 because phi(52*2) = phi(52*2+1).

Cf. A000010, A001274, A275337.

Table[k = 1; While[EulerPhi[k n] != EulerPhi[k n + 1], k++]; k, {n, 5}] (* Michael De Vlieger, Jul 27 2016 *)

a(n) = {my(k = 1); while (eulerphi(k*n) != eulerphi(k*n+1), k++); k; }

a(A001274(n)) = 1.

cp's OEIS Frontend

A326403 Numbers k such that iphi(k) = iphi(k+1), where iphi(k) is an infinitary analog to the Euler totient function (A091732).

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A057000 a(n) = phi(n+1) - phi(n).

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A001837 Numbers k such that phi(2k+1) < phi(2k).

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

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A067143 Numbers n such that phi(n+1) = 3*phi(n).

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A172314 Numbers k such that phi(k+1) = 4*phi(k).

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A266276 a(n) is the smallest number k such that phi(k) = n*phi(k-1).

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A275412 Least k such that phi(kn) = phi(kn+1), or 0 if no such k exists.