A007366 -id:A007366 - cp's OEIS frontend

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

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

A060668 Numbers k such that phi(x) = k has exactly 5 solutions.

Original entry on oeis.org

8, 20, 220, 272, 300, 368, 416, 456, 500, 656, 732, 848, 876, 1092, 1160, 1212, 1236, 1328, 1376, 1424, 1568, 1624, 1716, 1808, 2144, 2244, 2336, 2420, 2460, 2480, 2528, 2556, 2768, 3056, 3080, 3252, 3320, 3344, 3536, 3560, 3612, 3728, 3732, 3900, 4016

Offset: 1

Robert G. Wilson v, Apr 18 2001

nonn

			8 = phi(15) = phi(16) = phi(20) = phi(24) = phi(30).

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).

Cf. A000010.

Number of solutions: A007617 (0), A007366 (2), A007367 (3), A060667 (4), this sequence (5), A060669 (6), A060670 (7), A060671 (8), A060672 (9), A060673 (10), A060674 (11), A060675 (12).

a = Table[ 0, {5000} ]; Do[ p = EulerPhi[ n ]; If[ p < 5001, a[[ p ]]++ ], {n, 1, 15000} ]; Select[ Range[ 5000 ], a[[ # ]] == 5 & ]

is(n)=sum(k=1,n,eulerphi(k)==n)==5 \\ Charles R Greathouse IV, Mar 03 2014

is(k) = invphiNum(k) == 5 \\ Amiram Eldar, Nov 17 2024, using Max Alekseyev's invphi.gp

A060670 Numbers k such that phi(x) = k has exactly 7 solutions.

Original entry on oeis.org

32, 132, 156, 544, 912, 924, 1012, 1044, 1140, 1452, 1464, 1472, 1476, 1572, 1664, 1764, 2076, 2100, 2232, 2424, 2580, 2624, 2652, 3096, 3248, 3336, 3444, 3660, 3996, 4488, 4776, 4840, 5060, 5316, 5412, 5696, 6504, 6516, 6540, 6612, 6660, 6780, 6996, 7116

Offset: 1

Robert G. Wilson v, Apr 18 2001

nonn

			32 = phi(51) = phi(64) = phi(68) = phi(80) = phi(96) = phi(102) = phi(120).

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).

Cf. A000010.

Number of solutions: A007617 (0), A007366 (2), A007367 (3), A060667 (4), A060668 (5), A060669 (6), this sequence (7), A060671 (8), A060672 (9), A060673 (10), A060674 (11), A060675 (12).

a = Table[ 0, {8000} ]; Do[ p = EulerPhi[ n ]; If[ p < 8001, a[ [ p ] ]++ ], {n, 1, 25000} ]; Select[ Range[ 8000 ], a[ [ # ] ] == 7 & ]

is(n)=sum(i=1,n,eulerphi(i)==n)==7 \\ Charles R Greathouse IV, Mar 03 2014

is(k) = invphiNum(k) == 7 \\ Amiram Eldar, Nov 17 2024, using Max Alekseyev's invphi.gp

A060674 Numbers k such that phi(x) = k has exactly 11 solutions.

Original entry on oeis.org

48, 512, 540, 1000, 1836, 2136, 2176, 2320, 2340, 3216, 3648, 3936, 4284, 4352, 4356, 4784, 5088, 5640, 5936, 6216, 6576, 6816, 7120, 7224, 7280, 7752, 8100, 8184, 8496, 8520, 8760, 9040, 9296, 9660, 9680, 9900, 9996, 10332, 10860, 11640, 11680, 11844

Offset: 1

Robert G. Wilson v, Apr 18 2001

nonn

			48 = phi(65) = phi(104) = phi(105) = phi(112) = phi(130) = phi(140) = phi(144) = phi(156) = phi(168) = phi(180) = phi(210).

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).

Cf. A000010.

Number of solutions: A007617 (0), A007366 (2), A007367 (3), A060667 (4), A060668 (5), A060669 (6), A060670 (7), A060671 (8), A060672 (9), A060673 (10), this sequence (11), A060675 (12).

a = Table[ 0, {12500} ]; Do[ p = EulerPhi[ n ]; If[ p < 12501, a[ [ p ] ]++ ], {n, 1, 50000} ]; Select[ Range[ 12500 ], a[ [ # ] ] == 11 & ]

is(n)=sum(i=1,n,eulerphi(i)==n)==11 \\ Charles R Greathouse IV, Mar 03 2014

is(k) = invphiNum(k) == 11 \\ Amiram Eldar, Nov 17 2024, using Max Alekseyev's invphi.gp

A060667 Numbers k such that phi(x) = k has exactly 4 solutions.

Original entry on oeis.org

4, 6, 18, 42, 100, 162, 184, 208, 328, 424, 460, 468, 486, 492, 616, 636, 664, 688, 700, 712, 784, 820, 900, 904, 1020, 1060, 1072, 1168, 1240, 1264, 1276, 1288, 1300, 1356, 1360, 1384, 1404, 1458, 1480, 1528, 1672, 1740, 1768, 1864, 1896, 1900, 1908, 2008

Offset: 1

Robert G. Wilson v, Apr 18 2001

nonn

			18 = phi(19) = phi(27) = phi(38) = phi(54).

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).

Cf. A000010.

Number of solutions: A007617 (0), A007366 (2), A007367 (3), this sequence (4), A060668 (5), A060669 (6), A060670 (7), A060671 (8), A060672 (9), A060673 (10), A060674 (11), A060675 (12).

a = Table[ 0, {2500} ]; Do[ p = EulerPhi[ n ]; If[ p < 2501, a[ [ p ] ]++ ], {n, 1, 5000} ]; Select[ Range[ 2500 ], a[ [ # ] ] == 4 & ]

is(k) = invphiNum(k) == 4 \\ Amiram Eldar, Nov 17 2024, using Max Alekseyev's invphi.gp

A060669 Numbers k such that phi(x) = k has exactly 6 solutions.

Original entry on oeis.org

12, 16, 84, 88, 112, 232, 348, 408, 592, 736, 760, 780, 832, 952, 984, 1032, 1048, 1068, 1128, 1232, 1272, 1312, 1332, 1428, 1432, 1488, 1552, 1608, 1692, 1912, 2052, 2200, 2272, 2292, 2436, 2484, 2552, 2576, 2608, 2632, 2700, 2728, 2832, 2848, 3048, 3088

Offset: 1

Robert G. Wilson v, Apr 18 2001

nonn

			12 = phi(13) = phi(21) = phi(26) = phi(28) = phi(36) = phi(42).

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).

Cf. A000010.

Number of solutions: A007617 (0), A007366 (2), A007367 (3), A060667 (4), A060668 (5), this sequence (6), A060670 (7), A060671 (8), A060672 (9), A060673 (10), A060674 (11), A060675 (12).

a = Table[ 0, {4000} ]; Do[ p = EulerPhi[ n ]; If[ p < 4001, a[ [ p ] ]++ ], {n, 1, 15000} ]; Select[ Range[ 4000 ], a[ [ # ] ] == 6 & ]
Take[Select[Tally[EulerPhi[Range[50000]]],#[[2]]==6&][[All,1]]//Sort,50] (* Harvey P. Dale, Sep 15 2016 *)

is(n)=sum(i=1,n,eulerphi(i)==n)==6 \\ Charles R Greathouse IV, Mar 03 2014

is(k) = invphiNum(k) == 6 \\ Amiram Eldar, Nov 17 2024, using Max Alekseyev's invphi.gp

A060671 Numbers k such that phi(x) = k has exactly 8 solutions.

Original entry on oeis.org

36, 64, 176, 200, 224, 280, 324, 464, 520, 888, 920, 1184, 1368, 1400, 1520, 1696, 1720, 1904, 1960, 2040, 2096, 2120, 2256, 2392, 2600, 2656, 2712, 2752, 2864, 2944, 2960, 2968, 2976, 2988, 3104, 3276, 3300, 3408, 3616, 3640, 3792, 3800, 3816, 3824, 3880

Offset: 1

Robert G. Wilson v, Apr 18 2001

nonn

			36 = phi(37) = phi(57) = phi(63) = phi(74) = phi(76) = phi(108) = phi(114) = phi(126).

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).

Cf. A000010.

Number of solutions: A007617 (0), A007366 (2), A007367 (3), A060667 (4), A060668 (5), A060669 (6), A060670 (7), this sequence (8), A060672 (9), A060673 (10), A060674 (11), A060675 (12).

a = Table[ 0, {5000} ]; Do[ p = EulerPhi[ n ]; If[ p < 5001, a[ [ p ] ]++ ], {n, 1, 25000} ]; Select[ Range[ 5000 ], a[ [ # ] ] == 8 & ]

is(n)=sum(i=1,n,eulerphi(i)==n)==8 \\ Charles R Greathouse IV, Mar 03 2014

is(k) = invphiNum(k) == 8 \\ Amiram Eldar, Nov 17 2024, using Max Alekseyev's invphi.gp

A060672 Numbers k such that phi(x) = k has exactly 9 solutions.

Original entry on oeis.org

40, 60, 108, 128, 252, 276, 440, 612, 696, 996, 1088, 1380, 1500, 1824, 1860, 1932, 2064, 2472, 2796, 2928, 3060, 3132, 3384, 3516, 4584, 4932, 5076, 5136, 5436, 5700, 5888, 6096, 6372, 6640, 6744, 7020, 7080, 7380, 7452, 7476, 7704, 8040, 8676, 8892

Offset: 1

Robert G. Wilson v, Apr 18 2001

nonn

			40 = phi(41) = phi(55) = phi(75) = phi(82) = phi(88) = phi(100) = phi(110) = phi(132) = phi(150).

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).

Cf. A000010.

Number of solutions: A007617 (0), A007366 (2), A007367 (3), A060667 (4), A060668 (5), A060669 (6), A060670 (7), A060671 (8), this sequence (9), A060673 (10), A060674 (11), A060675 (12).

a = Table[ 0, {10000} ]; Do[ p = EulerPhi[ n ]; If[ p < 10001, a[ [ p ] ]++ ], {n, 1, 35000} ]; Select[ Range[ 10000 ], a[ [ # ] ] == 9 & ]

is(n)=sum(i=1,n,eulerphi(i)==n)==9 \\ Charles R Greathouse IV, Mar 03 2014

is(k) = invphiNum(k) == 9 \\ Amiram Eldar, Nov 17 2024, using Max Alekseyev's invphi.gp

A060673 Numbers k such that phi(x) = k has exactly 10 solutions.

Original entry on oeis.org

24, 80, 180, 256, 264, 828, 1188, 1640, 1968, 2024, 2368, 2544, 2720, 2772, 2904, 3036, 3136, 3144, 3328, 3392, 3420, 4192, 4392, 4464, 4600, 5000, 5312, 5504, 5508, 5688, 5728, 5796, 5800, 6208, 6228, 6732, 6888, 7000, 7232, 7740, 7956, 8388, 8576, 9088

Offset: 1

Robert G. Wilson v, Apr 18 2001

nonn

			24 = phi(35) = phi(39) = phi(45) = phi(52) = phi(56) = phi(70) = phi(72) = phi(78) = phi(84) = phi(90).

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).

Cf. A000010.

Number of solutions: A007617 (0), A007366 (2), A007367 (3), A060667 (4), A060668 (5), A060669 (6), A060670 (7), A060671 (8), A060672 (9), this sequence (10), A060674 (11), A060675 (12).

a = Table[ 0, {10000} ]; Do[ p = EulerPhi[ n ]; If[ p < 10001, a[ [ p ] ]++ ], {n, 1, 35000} ]; Select[ Range[ 10000 ], a[ [ # ] ] == 10 & ]
Take[Select[Tally[EulerPhi[Range[50000]]],#[[2]]==10&][[;;,1]]//Union,50] (* Harvey P. Dale, Sep 15 2023 *)

is(n)=sum(i=1,n,eulerphi(i)==n)==10 \\ Charles R Greathouse IV, Mar 03 2014

is(k) = invphiNum(k) == 10 \\ Amiram Eldar, Nov 17 2024, using Max Alekseyev's invphi.gp

A060675 Numbers k such that phi(x) = k has exactly 12 solutions.

Original entry on oeis.org

160, 168, 352, 448, 816, 928, 972, 1024, 1176, 1848, 2464, 3040, 3808, 4152, 4440, 4512, 4736, 4944, 5104, 5152, 5160, 5304, 5952, 6408, 6656, 6672, 6784, 7648, 8384, 8704, 8904, 10432, 10528, 10624, 11000, 11008, 11456, 11776, 12048, 12416, 13024, 13032

Offset: 1

Robert G. Wilson v, Apr 18 2001

nonn

			160 = phi(187) = phi(205) = phi(328) = phi(352) = phi(374) = phi(400) = phi(410) = phi(440) = phi(492) = phi(528) = phi(600) = phi(660).

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).

Cf. A000010.

Number of solutions: A007617 (0), A007366 (2), A007367 (3), A060667 (4), A060668 (5), A060669 (6), A060670 (7), A060671 (8), A060672 (9), A060673 (10), A060674 (11), this sequence (12).

a = Table[ 0, {15000} ]; Do[ p = EulerPhi[ n ]; If[ p < 15001, a[ [ p ] ]++ ], {n, 1, 60000} ]; Select[ Range[ 15000 ], a[ [ # ] ] == 12 & ]
Union[Transpose[Select[Tally[EulerPhi[Range[100000]]],#[[2]]==12&]][[1]]] (* Harvey P. Dale, Oct 05 2015 *)

is(n)=sum(i=1,n,eulerphi(i)==n)==12 \\ Charles R Greathouse IV, Mar 03 2014

is(k) = invphiNum(k) == 12 \\ Amiram Eldar, Nov 17 2024, using Max Alekseyev's invphi.gp

cp's OEIS Frontend

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

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A060668 Numbers k such that phi(x) = k has exactly 5 solutions.

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A060670 Numbers k such that phi(x) = k has exactly 7 solutions.

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A060674 Numbers k such that phi(x) = k has exactly 11 solutions.

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A060667 Numbers k such that phi(x) = k has exactly 4 solutions.

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A060669 Numbers k such that phi(x) = k has exactly 6 solutions.

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A060671 Numbers k such that phi(x) = k has exactly 8 solutions.

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A060672 Numbers k such that phi(x) = k has exactly 9 solutions.