A050473 -id:A050473 - 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

A050472 Numbers m such that 2*phi(m) = phi(m+1).

Original entry on oeis.org

2, 4, 16, 154, 256, 286, 364, 804, 1066, 2146, 3382, 4550, 6106, 7700, 8176, 9268, 11284, 12556, 12970, 16402, 19228, 19276, 20272, 25132, 26404, 27346, 29154, 29574, 35644, 36418, 38368, 39646, 40494, 47214, 52234, 54652, 65536, 84862

Offset: 1

Jud McCranie, Dec 24 1999

nonn

			phi(256)=128, phi(256+1)=2*128, so 256 is a member of the sequence.

R. K. Guy, Unsolved Problems Number Theory, Sect. B36.

Giovanni Resta, Table of n, a(n) for n = 1..10000 (first 1000 terms from Donovan Johnson)

Cf. A000010, A001274, A050473, A171271.

Select[Range[85000], 2EulerPhi[#]==EulerPhi[#+1] &] (* Stefano Spezia, Apr 07 2025 *)

isok(n) = 2*eulerphi(n) == eulerphi(n+1); \\ Michel Marcus, Aug 02 2018

Conjecture : a(n)/n^3 is bounded. Does lim n -> infinity a(n)/n^3 = 2 ? - Benoit Cloitre, Aug 07 2002

a(n) = A171271(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

A110179 Least k such that phi(n+k) = 2*phi(n), where phi is Euler's totient function.

Original entry on oeis.org

2, 1, 2, 1, 10, 2, 6, 7, 4, 5, 14, 3, 22, 7, 2, 1, 34, 3, 18, 12, 14, 3, 46, 8, 16, 9, 10, 7, 58, 2, 30, 19, 8, 17, 30, 3, 36, 19, 26, 11, 82, 3, 86, 11, 20, 23, 94, 3, 80, 5, 34, 13, 106, 3, 68, 9, 16, 29, 118, 4, 82, 15, 10, 21, 32, 9, 94, 17, 20, 34, 142, 32, 112, 17, 48, 15, 66, 26

Offset: 1

T. D. Noe, Jul 15 2005

nonn

Makowski shows that a k exists for each n. It appears that k <= 2n. For prime n, it appears that n-1 <= k <= 2n.

Richard K. Guy, Unsolved Problems in Number Theory, 3rd Ed., New York, Springer-Verlag, 2004, Section B36, p. 138.

Donovan Johnson, 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, A050473 (least k such that phi(n+k) = 2*phi(k)).

Table[k=1; e=EulerPhi[n]; While[EulerPhi[n+k] != 2e, k++ ]; k, {n, 100}]

a(n) = vecmin(select(x -> x > n, invphi(2*eulerphi(n)))) - n; \\ Amiram Eldar, Nov 05 2024, using Max Alekseyev's invphi.gp

A110196 Numbers m such that k = 2m is the least k such that phi(m+k) = 2*phi(k).

Original entry on oeis.org

1, 13, 23, 97, 113, 131, 199, 227, 491, 859, 929

Offset: 1

T. D. Noe, Jul 15 2005

Note that all m > 1 are primes.

No other terms below 10^8. - Max Alekseyev, Nov 18 2024

Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).

Cf. A000010, A050473 (least k such that phi(n+k) = 2*phi(k)).

Do[k=1; While[EulerPhi[n+k] != 2*EulerPhi[k], k++ ]; If[k==2n, Print[n]], {n, 5000}]

f(n) = apply(x -> x - n, select(x -> x > n, invphi(2*eulerphi(n)))); \\ using Max Alekseyev's invphi.gp
lista(nmax) = {my(v = vector(nmax), c = 0, k = 1, s); while(c < nmax, s = f(k); for(i = 1, #s, if(s[i] <= nmax && v[s[i]] == 0, c++; v[s[i]] = k)); k++); for(i = 1, #v, if(v[i] == 2*i, print1(i, ", ")));} \\ Amiram Eldar, Nov 05 2024

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

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A050472 Numbers m such that 2*phi(m) = phi(m+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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A110179 Least k such that phi(n+k) = 2*phi(n), where phi is Euler's totient function.

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A110196 Numbers m such that k = 2m is the least k such that phi(m+k) = 2*phi(k).

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