A007617 -id:A007617 - cp's OEIS frontend

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

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

A058812 Irregular triangle of rows of numbers in increasing order. Row 1 = {1}. Row m + 1 contains all numbers k such that phi(k) is in row m.

Original entry on oeis.org

1, 2, 3, 4, 6, 5, 7, 8, 9, 10, 12, 14, 18, 11, 13, 15, 16, 19, 20, 21, 22, 24, 26, 27, 28, 30, 36, 38, 42, 54, 17, 23, 25, 29, 31, 32, 33, 34, 35, 37, 39, 40, 43, 44, 45, 46, 48, 49, 50, 52, 56, 57, 58, 60, 62, 63, 66, 70, 72, 74, 76, 78, 81, 84, 86, 90, 98, 108, 114, 126

Offset: 0

Labos Elemer, Jan 03 2001

Nontotient values (A007617) are also present as inverses of some previous value.

Old name was: Irregular triangle of inverse totient values of integers generated recursively. Initial value is 1. The inverse-phi sets in increasing order are as follows: {1} -> {2} -> {3, 4, 6} -> {5, 7, 8, 9, 10, 12, 14, 18} -> ... The terms of each row are arranged by magnitude. The next row starts when the increase of terms is violated. 2^n is included in the n-th row. - David A. Corneth, Mar 26 2019

			Triangle begins:
  1;
  2;
  3, 4, 6;
  5, 7, 8, 9, 10, 12, 14, 18;
  ...
Row 3 is {3, 4, 6} as for each number k in this row, phi(k) is in row 2. - _David A. Corneth_, Mar 26 2019

T. D. Noe, Rows n=0..9 of triangle, flattened
Hartosh Singh Bal, Gaurav Bhatnagar, Prime number conjectures from the Shapiro class structure, arXiv:1903.09619 [math.NT], 2019.
T. D. Noe, Primes in classes of the iterated totient function, JIS 11 (2008) 08.1.2.
Index entries for sequences that are permutations of the natural numbers

Cf. A000010, A007617, A005277.
A058811 gives the number of terms in each row.
Cf. A003434, A007755, A060611, A092878, A098196, A135832.
Cf. also A334111.

inversePhi[m_?OddQ] = {}; inversePhi[1] = {1, 2}; inversePhi[m_] := Module[{p, nmax, n, nn}, p = Select[Divisors[m] + 1, PrimeQ]; nmax = m*Times @@ (p/(p-1)); n = m; nn = {}; While[n <= nmax, If[EulerPhi[n] == m, AppendTo[nn, n]]; n++]; nn]; row[n_] := row[n] = inversePhi /@ row[n-1] // Flatten // Union; row[0] = {1}; row[1] = {2}; Table[row[n], {n, 0, 5}] // Flatten (* Jean-François Alcover, Dec 06 2012 *)

Definition revised by T. D. Noe, Nov 30 2007

New name from David A. Corneth, Mar 26 2019

A066678 Totients of the least numbers for which the totient is divisible by n.

Original entry on oeis.org

1, 2, 6, 4, 10, 6, 28, 8, 18, 10, 22, 12, 52, 28, 30, 16, 102, 18, 190, 20, 42, 22, 46, 24, 100, 52, 54, 28, 58, 30, 310, 32, 66, 102, 70, 36, 148, 190, 78, 40, 82, 42, 172, 44, 180, 46, 282, 48, 196, 100, 102, 52, 106, 54, 110, 56, 228, 58, 708, 60, 366, 310, 126, 64

Offset: 1

Labos Elemer, Dec 22 2001

nonn

From Alonso del Arte, Feb 03 2017: (Start)
One of the less obvious consequences of Dirichlet's theorem on primes in arithmetic progression is that this sequence is well-defined for all positive integers.
Suppose n is a nontotient (see A007617). Obviously a(n) != n. Dirichlet's theorem assures us that, if nothing else, there are infinitely many primes of the form nk + 1 for k positive (and in this case, k > 1). Then phi(nk + 1) = nk, suggesting a(n) = nk corresponding to the smallest k.
Of course not all a(n) are 1 less than a prime, such as 8, 20, 24, 54, etc. (End)

			a(23) = 46 because there is no solution to phi(x) = 23 but there are solutions to phi(x) = 46, like x = 47.
a(24) = 24 because there are solutions to phi(x) = 24, such as x = 35.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..5000 from Vincenzo Librandi)

Cf. A000010, A066674, A066675, A066676, A066677, A067005, A061026.

EulerPhi[mulTotientList = ConstantArray[1, 70]; k = 1; While[Length[vac = Rest[Flatten[Position[mulTotientList, 1]]]] > 0, k++; mulTotientList[[Intersection[Divisors[EulerPhi[k]], vac]]] *= k]; mulTotientList] (* Vincenzo Librandi Feb 04 2017 *)
a[n_] := For[k=1, True, k++, If[Divisible[t = EulerPhi[k], n], Return[t]]];
Array[a, 64] (* Jean-François Alcover, Jul 30 2018 *)

list(len) = {my(v = vector(len), c = 0, k = 1, e); while(c < len, e = eulerphi(k); fordiv(e, d, if(d <= len && v[d] == 0, v[d] = e; c++)); k++); v;} \\ Amiram Eldar, Mar 07 2025

def A066678(n):
    s = 1
    while euler_phi(s) % n: s += 1
    return euler_phi(s)
print([A066678(n) for n in (1..64)]) # Peter Luschny, Feb 05 2017

a(n) = A000010(A061026(n)).

cp's OEIS Frontend

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.

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

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

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