A050472 -id:A050472 - cp's OEIS frontend

A001838 Numbers k such that phi(k+2) = phi(k) + 2.

3, 5, 6, 11, 12, 14, 17, 18, 20, 29, 41, 44, 59, 62, 71, 92, 101, 107, 116, 137, 149, 164, 179, 191, 197, 212, 227, 239, 254, 269, 281, 311, 332, 347, 356, 419, 431, 452, 461, 521, 524, 569, 599, 617, 641, 659, 692, 716, 764, 809, 821, 827, 857, 881, 932, 956

Offset: 1

N. J. A. Sloane

If p and p+2 are primes then p is a solution. If p and 2p+1 are both odd primes then 4p is a solution. Several numbers of the form 2^j-2 are solutions (see cross-referenced sequences). Although 18 is a solution, it is not of any of these forms.

Twice Mersenne primes (cf. A000668) are also solutions. - Vladeta Jovovic, Feb 14 2002

			phi(18+2) = 8 = phi(18) + 2, so 18 is in the sequence.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 840.
D. M. Burton, Elementary Number Theory, section 7-2.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence as N0951, although there are errors, probably caused by errors in the original source).
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..10000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
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.
L. Moser, Some equations involving Euler's totient function, Amer. Math. Monthly, 56 (1949), 22-23.

Cf. A050472, A050473, etc. Essentially the same as A056853.

import Data.List (elemIndices)
a001838 n = a001838_list !! (n-1)
a001838_list = map (+ 1) $ elemIndices 2 $
   zipWith (-) (drop 2 a000010_list) a000010_list
-- Reinhard Zumkeller, Feb 21 2012

[n: n in [1..1000] | EulerPhi(n+2) eq EulerPhi(n)+2]; // Vincenzo Librandi, Sep 11 2015

Select[Range@1000, EulerPhi@(# + 2)== EulerPhi[#] + 2 &] (* Vincenzo Librandi, Sep 11 2015 *)
Position[Partition[EulerPhi[Range[1000]],3,1],?(#[[1]]+2 == #[[3]]&), 1, Heads->False]//Flatten (* _Harvey P. Dale, Oct 04 2017 *)

isok(n) = eulerphi(n+2) == eulerphi(n) + 2; \\ Michel Marcus, Sep 11 2015

More terms from Jud McCranie, Dec 24 1999

A171271 Numbers n such that phi(n)=2*phi(n-1).

Original entry on oeis.org

3, 5, 17, 155, 257, 287, 365, 805, 1067, 2147, 3383, 4551, 6107, 7701, 8177, 9269, 11285, 12557, 12971, 16403, 19229, 19277, 20273, 25133, 26405, 27347, 29155, 29575, 35645, 36419, 38369, 39647, 40495, 47215, 52235, 54653, 65537, 84863

Offset: 1

Farideh Firoozbakht, Feb 23 2010

Theorem: A prime p is in the sequence iff p is a Fermat prime.

Proof: If p=2^2^n+1 is prime (Fermat prime) then phi(p)=2^2^n=2* phi(2^2^n)=2*phi(p-1), so p is in the sequence. Now if p is a prime term of the sequence then phi(p)=2*phi(p-1) so p-1=2*phi(p-1) and we deduce that p-1=2^m hence p is a Fermat prime.

Giovanni Resta, Table of n, a(n) for n = 1..10000 (first 5416 terms from Hiroaki Yamanouchi)

Cf. A019434, A050472, A171262.

[n: n in [2..2*10^5] | EulerPhi(n) eq 2*EulerPhi(n-1)]; // Vincenzo Librandi, May 17 2015

Select[Range[85000],EulerPhi[ # ]==2EulerPhi[ #-1]&]
Flatten[Position[Partition[EulerPhi[Range[90000]],2,1],?(2#[[1]] == #[[2]]&),1,Heads->False]]+1 (* _Harvey P. Dale, Sep 09 2017 *)

a(n) = A050472(n) + 1. - Ray Chandler, May 01 2015

A055458 a(n) = smallest composite solution x to the equation phi(x+2n) = phi(x)+2n.

Original entry on oeis.org

6, 12, 21, 24, 36, 45, 48, 39, 63, 72, 72, 95, 60, 57, 224, 84, 15, 135, 1058, 45, 301, 144

Offset: 1

Labos Elemer, Jun 26 2000

Sivaramakrishnan (1989) quotes Makowski, who gave solutions for phi(x+d) = phi(x)+d with d = 2^a and d = 2*3^a. Compare also A007694 and A049237.
Smallest prime solutions appear to be identical with A054906.
a(23) is presently unknown.
The sequence continues as (with ? for unknown values): ?, 95, 162, 63, 189, 69, 156, 161, 180, 69, 260, 150, ?, 115, 204, 129, 400, 75, 180, 165, 35, 117, 476, 7105, 288, 195, ?, 324, 620, 240, 81, 145, 14531, 153, 644, 309, ?, 203, ?, 63, 640, 75, 372, 285, 2312, 33, 343, 642, 336, 155, ?, 147, 728, 396, 1564, 185, 564, 87, 567, 360, 360, 155, 492, 510, 560, 516, 516, 301, 4232, 261, 860, 387, 576, 185, 564, 309, 1000, 225 ... - Don Reble, Apr 29 2015

			a(19) = 1058 because phi(1058 + 38) = phi(1096) = 544 = 506 + 38 = phi(1058) + 38.
a(100) = 225, phi(225 + 200) = phi(425) = 320 = 120 + 200 = phi(225) + 200.

Sivaramakrishnan, R. (1989): Classical theory of Arithmetical Functions. Marcel Dekker, Inc., New York-Basel. Chapter V, Problem 20, page 113.

Cf. A000010, A054906, A050472, A050473, A007694, A049237.

A055458 := proc(n)
    local x;
    for x from 0 do
        if not isprime(x) then
        if numtheory[phi](x+2*n) = numtheory[phi](x)+2*n then
            return x;
        end if;
        end if;
    end do:
end proc: # R. J. Mathar, Sep 23 2016

Table[k = 4; While[Nand[CompositeQ@ k, EulerPhi[k + 2 n] == EulerPhi[k] + 2 n], k++]; k, {n, 22}] (* Michael De Vlieger, Dec 17 2016 *)

a(n)=forcomposite(x=4,, if(eulerphi(x+2*n) == eulerphi(x)+2*n, return(x))) \\ does not handle -1s; Charles R Greathouse IV, Apr 28 2015

More terms from Michel ten Voorde Jun 14 2003

Entry revised by N. J. A. Sloane, Apr 28 2015

A050473 Smallest k such that phi(k+n) = 2*phi(k).

Original entry on oeis.org

2, 1, 1, 2, 1, 4, 3, 4, 3, 5, 5, 8, 26, 7, 5, 8, 9, 12, 5, 10, 7, 8, 46, 16, 5, 13, 9, 14, 7, 25, 21, 13, 9, 17, 7, 24, 62, 19, 11, 20, 76, 28, 13, 16, 15, 23, 17, 32, 21, 25, 17, 26, 52, 36, 11, 28, 13, 26, 13, 45, 74, 28, 17, 26, 13, 39, 33, 31, 21, 32, 13, 48, 39, 37, 25, 38

Offset: 1

Jud McCranie, Dec 24 1999

Makowski proved that the sequence is well-defined.

It appears that k <= 2n, with equality for the n in A110196 only. Computations for n < 10^6 appear to show that k < n for all but a finite number of n. - T. D. Noe, Jul 15 2005

			phi(13+26) = 24 = 2*phi(13), so a(13) = 26.

Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B36, p. 138.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).
Andrzej Makowski, On the equation phi(n+k)=2*phi(n), Elem. Math., Vol. 29, No. 1 (1974), p. 13.

Cf. A000010, A007015, A050472, A110196.

Cf. A110179 (least k such that phi(n+k)=2*phi(n)).

Table[k=1; While[EulerPhi[n+k] != 2*EulerPhi[k], k++ ]; k, {n, 100}] (Noe)

f(n) = apply(x -> x - n, select(x -> x > n, invphi(2*eulerphi(n)))); \\ using Max Alekseyev's invphi.gp
lista(len) = {my(v = vector(len), c = 0, k = 1, s); while(c < len, s = f(k); for(i = 1, #s, if(s[i] <= len && v[s[i]] == 0, c++; v[s[i]] = k)); k++); v;} \\ Amiram Eldar, Nov 05 2024

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

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

cp's OEIS Frontend

A001838 Numbers k such that phi(k+2) = phi(k) + 2.

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A171271 Numbers n such that phi(n)=2*phi(n-1).

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A055458 a(n) = smallest composite solution x to the equation phi(x+2n) = phi(x)+2n.

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A050473 Smallest k such that phi(k+n) = 2*phi(k).

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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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