cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 13 results. Next

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

Views

Author

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

Keywords

Comments

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

Examples

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

References

  • 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).

Links

Crossrefs

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).

Programs

  • Maple
    select(nops@numtheory:-invphi=2, [$1..1000]); # Robert Israel, Dec 20 2017
  • Mathematica
    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 *)
  • PARI
    is(k) = invphiNum(k) == 2 \\ Amiram Eldar, Nov 16 2024, using Max Alekseyev's invphi.gp

Formula

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

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

Views

Author

Robert G. Wilson v, Apr 18 2001

Keywords

Examples

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

Links

Crossrefs

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).

Programs

  • Mathematica
    a = Table[ 0, {5000} ]; Do[ p = EulerPhi[ n ]; If[ p < 5001, a[[ p ]]++ ], {n, 1, 15000} ]; Select[ Range[ 5000 ], a[[ # ]] == 5 & ]
  • PARI
    is(n)=sum(k=1,n,eulerphi(k)==n)==5 \\ Charles R Greathouse IV, Mar 03 2014
  • PARI
    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

Views

Author

Robert G. Wilson v, Apr 18 2001

Keywords

Examples

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

Links

Crossrefs

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).

Programs

  • Mathematica
    a = Table[ 0, {8000} ]; Do[ p = EulerPhi[ n ]; If[ p < 8001, a[ [ p ] ]++ ], {n, 1, 25000} ]; Select[ Range[ 8000 ], a[ [ # ] ] == 7 & ]
  • PARI
    is(n)=sum(i=1,n,eulerphi(i)==n)==7 \\ Charles R Greathouse IV, Mar 03 2014
  • PARI
    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

Views

Author

Robert G. Wilson v, Apr 18 2001

Keywords

Examples

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

Links

Crossrefs

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).

Programs

  • Mathematica
    a = Table[ 0, {12500} ]; Do[ p = EulerPhi[ n ]; If[ p < 12501, a[ [ p ] ]++ ], {n, 1, 50000} ]; Select[ Range[ 12500 ], a[ [ # ] ] == 11 & ]
  • PARI
    is(n)=sum(i=1,n,eulerphi(i)==n)==11 \\ Charles R Greathouse IV, Mar 03 2014
  • PARI
    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

Views

Author

Robert G. Wilson v, Apr 18 2001

Keywords

Examples

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

Links

Crossrefs

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).

Programs

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

Views

Author

Robert G. Wilson v, Apr 18 2001

Keywords

Examples

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

Links

Crossrefs

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).

Programs

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

Views

Author

Robert G. Wilson v, Apr 18 2001

Keywords

Examples

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

Links

Crossrefs

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).

Programs

  • Mathematica
    a = Table[ 0, {5000} ]; Do[ p = EulerPhi[ n ]; If[ p < 5001, a[ [ p ] ]++ ], {n, 1, 25000} ]; Select[ Range[ 5000 ], a[ [ # ] ] == 8 & ]
  • PARI
    is(n)=sum(i=1,n,eulerphi(i)==n)==8 \\ Charles R Greathouse IV, Mar 03 2014
  • PARI
    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

Views

Author

Robert G. Wilson v, Apr 18 2001

Keywords

Examples

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

Links

Crossrefs

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).

Programs

  • Mathematica
    a = Table[ 0, {10000} ]; Do[ p = EulerPhi[ n ]; If[ p < 10001, a[ [ p ] ]++ ], {n, 1, 35000} ]; Select[ Range[ 10000 ], a[ [ # ] ] == 9 & ]
  • PARI
    is(n)=sum(i=1,n,eulerphi(i)==n)==9 \\ Charles R Greathouse IV, Mar 03 2014
  • PARI
    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

Views

Author

Robert G. Wilson v, Apr 18 2001

Keywords

Examples

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

Links

Crossrefs

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).

Programs

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

Views

Author

Robert G. Wilson v, Apr 18 2001

Keywords

Examples

			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).

Links

Crossrefs

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).

Programs

  • Mathematica
    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 *)
  • PARI
    is(n)=sum(i=1,n,eulerphi(i)==n)==12 \\ Charles R Greathouse IV, Mar 03 2014
  • PARI
    is(k) = invphiNum(k) == 12 \\ Amiram Eldar, Nov 17 2024, using Max Alekseyev's invphi.gp
Showing 1-10 of 13 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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