A007015 -id:A007015 - cp's OEIS frontend

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

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

A007366 Numbers k such that phi(x) = k has exactly 2 solutions.

Original entry on oeis.org

1, 10, 22, 28, 30, 46, 52, 54, 58, 66, 70, 78, 82, 102, 106, 110, 126, 130, 136, 138, 148, 150, 166, 172, 178, 190, 196, 198, 210, 222, 226, 228, 238, 250, 262, 268, 270, 282, 292, 294, 306, 310, 316, 330, 342, 346, 358, 366, 372, 378, 382, 388, 418, 430, 438

Offset: 1

N. J. A. Sloane, Mira Bernstein, Robert G. Wilson v

nonn

Contains {2*3^(6k+1): k >= 1} as a subsequence. This is the simplest proof for the infinity of these numbers (see Sierpiński, Exercise 12, p. 237). - Franz Vrabec, Aug 21 2021

The smaller of the solutions to phi(x) = a(n) is given by A271983(n). It is conjectured that the larger solution is 2*A271983(n); or equivalently, all terms in A271983 are odd. - Jianing Song, Nov 08 2022

			10 = phi(11) = phi(22).

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.
Wacław Sierpiński, Elementary Theory of Numbers, Warszawa, 1964.
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..10000 (terms 1..1000 from T. D. Noe)
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].
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).
Robert G. Wilson v, Letter to N. J. A. Sloane, Jul. 1992.

Cf. A000010, A001221, A023900, A271983.

Number of solutions: A007617 (0), this sequence (2), A007367 (3), A060667 (4), A060668 (5), A060669 (6), A060670 (7), A060671 (8), A060672 (9), A060673 (10), A060674 (11), A060675 (12).

select(nops@numtheory:-invphi=2, [$1..1000]); # Robert Israel, Dec 20 2017

a = Table[ 0, {500} ]; Do[ p = EulerPhi[ n ]; If[ p < 501, a[ [ p ] ]++ ], {n, 1, 500} ]; Select[ Range[ 500 ], a[ [ # ] ] == 2 & ]
(* Second program: *)
With[{nn = 1325}, TakeWhile[Union@ Select[KeyValueMap[{#1, Length@ #2} &, PositionIndex@ Array[EulerPhi, nn]], Last@ # == 2 &][[All, 1]], # < nn/3 &] ] (* Michael De Vlieger, Dec 20 2017 *)

is(k) = invphiNum(k) == 2 \\ Amiram Eldar, Nov 16 2024, using Max Alekseyev's invphi.gp

#({phi^(-1)(a(n))}) = 2. - Torlach Rush, Dec 22 2017

A007367 Numbers k such that phi(x) = k has exactly 3 solutions.

Original entry on oeis.org

2, 44, 56, 92, 104, 116, 140, 164, 204, 212, 260, 296, 332, 344, 356, 380, 392, 444, 452, 476, 524, 536, 564, 584, 588, 620, 632, 684, 692, 716, 744, 764, 776, 836, 860, 884, 932, 956, 980, 1004, 1016, 1112, 1124, 1136, 1172, 1196, 1284, 1292, 1304

Offset: 1

N. J. A. Sloane, Mira Bernstein, Robert G. Wilson v

nonn

From Torlach Rush, Jul 23 2018: (Start)
For known terms:
- The greatest common divisor of the three solutions is the distance of the middle solution from the least solution and is half the distance of the middle solution to the largest solution.
- If the number of distinct prime factors of k equals the number of solutions of k = phi(x), then the greatest common divisor of the solutions is the least solution divided by the number of solutions.
- Except for a(1), if the largest prime factor is the same for all solutions and is equal to the greatest common divisor of all solutions then the distance from a(n) to the least solution is gcd({k: phi(k) = a(n)}) + 2.  (End)
By Ford's theorem on Euler totient function, this sequence is infinite. - Jianing Song, Jul 18 2018

			phi(69) = phi(92) = phi(138) = 44, so 44 is a term.

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.
Jean-Marie De Koninck, Ces nombres qui nous fascinent, Entry 44, p. 17, Ellipses, Paris, 2008.
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..10000 (terms 1..1000 from T. D. Noe)
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].
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).
Wikipedia, Ford's theorem.
Robert G. Wilson v, Letter to N. J. A. Sloane, Jul. 1992.

Cf. A000010, A002202, A058277, A085713.

Number of solutions: A007617 (0), A007366 (2), this sequence (3), A060667 (4), A060668 (5), A060669 (6), A060670 (7), A060671 (8), A060672 (9), A060673 (10), A060674 (11), A060675 (12).

a007367 n = a007367_list !! (n-1)
a007367_list = map fst $
               filter ((== 3) . snd) $ zip a002202_list a058277_list
-- Reinhard Zumkeller, Nov 25 2015

a = Table[ 0, {1500} ]; Do[ p = EulerPhi[ n ]; If[ p < 1501, a[ [ p ] ]++ ], {n, 1, 1500} ]; Select[ Range[ 1500 ], a[ [ # ] ] == 3 & ]
Take[Select[Tally[EulerPhi[Range[50000]]],#[[2]]==3&][[All,1]],50]//Sort (* Harvey P. Dale, Apr 02 2018 *)

is(k) = invphiNum(k) == 3 \\ Amiram Eldar, Nov 17 2024, using Max Alekseyev's invphi.gp

A007371 Numbers k such that sigma(x) = k has exactly 2 solutions.

Original entry on oeis.org

12, 18, 31, 32, 54, 56, 80, 98, 104, 108, 114, 124, 126, 128, 132, 140, 152, 156, 182, 186, 210, 264, 272, 280, 308, 320, 342, 378, 390, 392, 399, 403, 408, 416, 440, 444, 448, 492, 522, 532, 570, 572, 594, 608, 630, 632, 726, 762, 770, 774, 780, 784, 800

Offset: 1

N. J. A. Sloane, Mira Bernstein, Robert G. Wilson v

nonn

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.
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..10000 (terms 1..1000 from Donovan Johnson)
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].
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).
Robert G. Wilson v, Letter to N. J. A. Sloane, Jul. 1992.

Cf. A000203.

Number of solutions: A007369 (0), A007370 (1), this sequence (2), A007372 (3), A060660 (4), A060661 (5), A060662 (6), A060663 (7), A060664 (8), A060665 (9), A060666 (10), A060678 (11), A060676 (12).

a = Table[ 0, {750} ]; Do[ s = DivisorSigma[ 1, n ]; If[ s < 751, a[ [ s ] ]++ ], {n, 1, 750} ]; Select[ Range[ 750 ], a[ [ # ] ] == 2 & ]

is(n)=sum(k=1,n,sigma(k)==n)==2 \\ Charles R Greathouse IV, Mar 09 2014

is(k) = invsigmaNum(k) == 2 \\ Amiram Eldar, Nov 17 2024, using Max Alekseyev's invphi.gp

A007372 Numbers k such that sigma(x) = k has exactly 3 solutions.

Original entry on oeis.org

24, 42, 48, 60, 84, 90, 224, 228, 234, 248, 270, 294, 324, 450, 468, 528, 558, 620, 640, 660, 810, 882, 888, 896, 968, 972, 1020, 1050, 1104, 1116, 1140, 1216, 1232, 1240, 1274, 1332, 1392, 1400, 1452, 1456, 1464, 1482, 1524, 1530, 1600, 1694, 1716, 1760

Offset: 1

N. J. A. Sloane, Mira Bernstein, Robert G. Wilson v

nonn

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.
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..10000 (terms 1..1000 from Donovan Johnson)
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].
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).
Robert G. Wilson v, Letter to N. J. A. Sloane, Jul. 1992.

Cf. A000203.

Number of solutions: A007369 (0), A007370 (1), A007371 (2), this sequence (3), A060660 (4), A060661 (5), A060662 (6), A060663 (7), A060664 (8), A060665 (9), A060666 (10), A060678 (11), A060676 (12).

a = Table[ 0, {2500} ]; Do[ s = DivisorSigma[ 1, n ]; If[ s < 2501, a[ [ s ] ]++ ], {n, 1, 2500} ]; Select[ Range[ 2500 ], a[ [ # ] ] == 3 & ]

is(n)=sum(k=1,n,sigma(k)==n)==3 \\ Charles R Greathouse IV, Mar 09 2014

is(k) = invsigmaNum(k) == 3 \\ Amiram Eldar, Nov 17 2024, using Max Alekseyev's invphi.gp

A179188 Numbers n such that phi(n) = phi(n+6), with Euler's totient function phi=A000010.

Original entry on oeis.org

24, 34, 36, 39, 43, 44, 57, 72, 78, 82, 84, 93, 96, 108, 111, 146, 178, 201, 216, 222, 225, 226, 228, 306, 327, 364, 366, 381, 399, 417, 432, 438, 442, 466, 471, 482, 516, 527, 540, 543, 562, 576, 597, 610, 626, 633, 648, 706, 714, 732, 738, 802, 818, 866, 898, 912, 921, 924, 942, 948, 972, 1011

Offset: 1

M. F. Hasler, Jan 05 2011

nonn

There are 1385502728 terms under 10^12. - Jud McCranie, Feb 13 2012

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, A179186, A179187, A007015.

[n: n in [1..1000] | EulerPhi(n) eq EulerPhi(n+6)]; // Vincenzo Librandi, Sep 08 2016

Flatten[Position[Partition[EulerPhi[Range[1200]],7,1],?(#[[1]] == #[[7]]&),{1},Heads->False]] (* _Harvey P. Dale, Jan 30 2016 *)
Select[Range[1000], EulerPhi[#] == EulerPhi[# + 6] &] (* Vincenzo Librandi, Sep 08 2016 *)

{op=vector(N=6); 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)+6).

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,",")))}

A217139 Numbers n such that phi(n) = phi(n+12), with Euler's totient function phi = A000010.

Original entry on oeis.org

48, 68, 72, 78, 86, 88, 114, 143, 144, 156, 157, 164, 168, 186, 192, 203, 216, 222, 247, 273, 292, 356, 402, 432, 444, 450, 452, 456, 612, 654, 728, 732, 762, 798, 834, 864, 876, 884, 932, 942, 964, 1032, 1054, 1080, 1086, 1124, 1147, 1152, 1194, 1209, 1220

Offset: 1

Michel Marcus, Sep 27 2012

Most of numbers n in this sequence are divisible by 2, and it appears that n/2 belongs to A179188. The other ones are listed in sequence A217141.

Proof of the comment: If n is even and not a multiple of 4 then phi(n)=phi(n/2). If n is a multiple of 4 then phi(n)=2 * phi(n/2). So when k is a multiple of 4 and phi(n)=phi(n+k), then phi(n/2)=phi(n/2+k/2). QED. This also applies to A179186, A179202. - Jud McCranie, Dec 30 2012

Jud McCranie, Table of n, a(n) for n = 1..10000
Kevin Ford, Solutions of phi(n)=phi(n+k) and sigma(n)=sigma(n+k), arXiv:2002.12155 [math.NT], 2020.

Cf. A000010, A179188, A217141, A007015.

[n: n in [1..3000] | EulerPhi(n) eq EulerPhi(n+12)]; // Vincenzo Librandi, Sep 08 2016

Select[Range[1, 5000], EulerPhi[#] == EulerPhi[# + 12] &] (* Vincenzo Librandi, Jun 24 2014 *)

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

A179187 Numbers n such that phi(n)=phi(n+5), with Euler's totient function phi=A000010.

Original entry on oeis.org

5, 9, 15, 21, 15556, 21016, 25930, 25935, 27027, 36304, 46683, 129675, 266128, 307923, 329175, 430348, 503139, 636400, 684411, 812170, 1014778, 1252713, 1777545, 1871788, 1892452, 1911987, 2622160, 2629930, 2731360, 2947035, 3397480, 4200100, 5451537

Offset: 1

M. F. Hasler, Jan 05 2011

nonn

There are only 43 terms below 10^7, and 1843 terms below 10^12. [Jud McCranie, Feb 13 2012]

Jud McCranie, Table of n, a(n) for n = 1..1843 (terms < 10^12)
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, A007015.

[n: n in [1..5000000] | EulerPhi(n) eq EulerPhi(n+5)]; // Vincenzo Librandi, Sep 08 2016

Select[Range[5000000], EulerPhi[#] == EulerPhi[# + 5] &] (* Vincenzo Librandi, Sep 08 2016 *)
Position[Partition[EulerPhi[Range[55*10^5]],6,1],?(#[[1]] == #[[6]]&), {1}, Heads->False]//Flatten (* _Harvey P. Dale, Nov 10 2016 *)

{op=vector(N=5); for( n=1,1e7,if( op[n%N+1]+0==op[n%N+1]=eulerphi(n),print1(n-N,",")))}

A000010(a(n)) = A000010(a(n)+5).

A007365 Smallest k such that sigma(n+k) = sigma(k).

Original entry on oeis.org

1, 14, 33, 382, 51, 6, 20, 10, 15, 14, 21, 28, 35, 182, 24, 26, 30, 142, 40, 34, 42, 20, 57, 135, 70, 30, 99, 42, 66, 406, 88, 56, 60, 54, 93, 24, 105, 248, 147, 44, 63, 30, 80, 435, 114, 52, 196, 310, 140, 40, 105, 92, 160, 66, 120, 140, 105, 88, 352, 154

Offset: 0

N. J. A. Sloane, Mira Bernstein, Robert G. Wilson v

nonn

If p > 3 is prime, a(p) <= 14*p. - Robert Israel, Feb 21 2020

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.
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 = 0..10000 (first 1001 terms from T. D. Noe)
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].
R. G. Wilson, V, Letter to N. J. A. Sloane, Jul. 1992.

Cf. A065932, A065933. sigma(x)=A000203(x) is the sum of the divisors of x.

N:= 1000: # to get all terms before the first with n + a(n) > N
S:= map(numtheory:-sigma, [$1..N]):
Res:= NULL:
found:= true:
for n from 1 while found do
found:= false;
for k from 1 to N-n do
   if S[k] = S[k+n] then
     Res:= Res, k; found:= true; break;
   fi
od;
od:
Res; # Robert Israel, Feb 21 2020

sk[n_]:=Module[{k=1},While[DivisorSigma[1,k]!=DivisorSigma[1,n+k], k++];k]; Array[sk,60,0] (* Harvey P. Dale, Oct 10 2012 *)

A007365(m)= {local(k,n); for(k=1,m,n=1; while(sigma(n)!=sigma(n+k), n++); print1(n,","))} \\ Klaus Brockhaus

cp's OEIS Frontend

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

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A007366 Numbers k such that phi(x) = k has exactly 2 solutions.

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A007367 Numbers k such that phi(x) = k has exactly 3 solutions.

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A007371 Numbers k such that sigma(x) = k has exactly 2 solutions.

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A007372 Numbers k such that sigma(x) = k has exactly 3 solutions.

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A179188 Numbers n such that phi(n) = phi(n+6), with Euler's totient function phi=A000010.

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Magma

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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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