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Search: a007366 -id:a007366
Displaying 1-10 of 15 results found. page 1 2
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A007367 Numbers k such that phi(x) = k has exactly 3 solutions.
(Formerly M2163) 		+10
14
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
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
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.
J.-M. 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).
LINKS
T. D. Noe, Table of n, a(n) for n = 1..1000
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].
Wikipedia, Ford's theorem
R. G. Wilson, V, Letter to N. J. A. Sloane, Jul. 1992
EXAMPLE
phi(69) = phi(92) = phi(138) = 44, so 44 is a term.
MATHEMATICA
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 *)
PROG
(Haskell)
a007367 n = a007367_list !! (n-1)
a007367_list = map fst $
filter ((== 3) . snd) $ zip a002202_list a058277_list
-- Reinhard Zumkeller, Nov 25 2015
CROSSREFS
Cf. A000010, A007366, A060667-A060671.
Cf. A002202, A058277, A085713.
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Mira Bernstein, Robert G. Wilson v
STATUS
approved
A071388 Numbers k such that the cardinality of the set of solutions to phi(x) = k is a prime. +10
4
1, 2, 8, 10, 20, 22, 28, 30, 32, 44, 46, 48, 52, 54, 56, 58, 66, 70, 72, 78, 82, 92, 96, 102, 104, 106, 110, 116, 120, 126, 130, 132, 136, 138, 140, 148, 150, 156, 164, 166, 172, 178, 190, 196, 198, 204, 210, 212, 216, 220, 222, 226, 228, 238, 240, 250, 260, 262 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
All terms except 1 are even. - Robert Israel, Mar 29 2020
LINKS
Robert Israel, Table of n, a(n) for n = 1..10000
EXAMPLE
InvPhi[48]={65,104,105,112,130,140,144,156,168,180,210} has 11 terms, so 48 is here.
MAPLE
filter:= n -> isprime(nops(numtheory:-invphi(n))):
select(filter, [$1..400]); # Robert Israel, Mar 29 2020
CROSSREFS
Cf. A000010, A007366, A007367, A014197, A058277, A060668, A060670, A060674, A063512, A071386-A071389.
KEYWORD
nonn
AUTHOR
Labos Elemer, May 23 2002
STATUS
approved
A060667 Numbers n such that phi(x) = n has exactly 4 solutions. +10
2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
LINKS
T. D. Noe, Table of n, a(n) for n=1..1000
EXAMPLE
18 = phi(19) = phi(27) = phi(38) = phi(54).
MATHEMATICA
a = Table[ 0, {2500} ]; Do[ p = EulerPhi[ n ]; If[ p < 2501, a[ [ p ] ]++ ], {n, 1, 5000} ]; Select[ Range[ 2500 ], a[ [ # ] ] == 4 & ]
CROSSREFS
Cf. A007366, A007367.
KEYWORD
nonn
AUTHOR
Robert G. Wilson v, Apr 18 2001
STATUS
approved
A060668 Numbers n such that phi(x) = n has exactly 5 solutions. +10
2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
LINKS
T. D. Noe, Table of n, a(n) for n = 1..1000
EXAMPLE
8 = phi(15) = phi(16) = phi(20) = phi(24) = phi(30).
MATHEMATICA
a = Table[ 0, {5000} ]; Do[ p = EulerPhi[ n ]; If[ p < 5001, a[[ p ]]++ ], {n, 1, 15000} ]; Select[ Range[ 5000 ], a[[ # ]] == 5 & ]
PROG
(PARI) is(n)=sum(k=1, n, eulerphi(k)==n)==5 \\ Charles R Greathouse IV, Mar 03 2014
CROSSREFS
Cf. A007366, A007367.
KEYWORD
nonn
AUTHOR
Robert G. Wilson v, Apr 18 2001
STATUS
approved
A060670 Numbers n such that phi(x) = n has exactly 7 solutions. +10
2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
LINKS
T. D. Noe, Table of n, a(n) for n = 1..1000
EXAMPLE
32 = phi(51) = phi(64) = phi(68) = phi(80) = phi(96) = phi(102) = phi(120).
MATHEMATICA
a = Table[ 0, {8000} ]; Do[ p = EulerPhi[ n ]; If[ p < 8001, a[ [ p ] ]++ ], {n, 1, 25000} ]; Select[ Range[ 8000 ], a[ [ # ] ] == 7 & ]
PROG
(PARI) is(n)=sum(i=1, n, eulerphi(i)==n)==7 \\ Charles R Greathouse IV, Mar 03 2014
CROSSREFS
Cf. A007366, A007367.
KEYWORD
nonn
AUTHOR
Robert G. Wilson v, Apr 18 2001
STATUS
approved
A060671 Numbers n such that phi(x) = n has exactly 8 solutions. +10
2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
LINKS
T. D. Noe, Table of n, a(n) for n = 1..1000
EXAMPLE
36 = phi(37) = phi(57) = phi(63) = phi(74) = phi(76) = phi(108) = phi(114) = phi(126).
MATHEMATICA
a = Table[ 0, {5000} ]; Do[ p = EulerPhi[ n ]; If[ p < 5001, a[ [ p ] ]++ ], {n, 1, 25000} ]; Select[ Range[ 5000 ], a[ [ # ] ] == 8 & ]
PROG
(PARI) is(n)=sum(i=1, n, eulerphi(i)==n)==8 \\ Charles R Greathouse IV, Mar 03 2014
CROSSREFS
Cf. A007366, A007367.
KEYWORD
nonn
AUTHOR
Robert G. Wilson v, Apr 18 2001
STATUS
approved
A060674 Numbers n such that phi(x) = n has exactly 11 solutions. +10
2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
LINKS
T. D. Noe, Table of n, a(n) for n = 1..1000
EXAMPLE
48 = phi(65) = phi(104) = phi(105) = phi(112) = phi(130) = phi(140) = phi(144) = phi(156) = phi(168) = phi(180) = phi(210).
MATHEMATICA
a = Table[ 0, {12500} ]; Do[ p = EulerPhi[ n ]; If[ p < 12501, a[ [ p ] ]++ ], {n, 1, 50000} ]; Select[ Range[ 12500 ], a[ [ # ] ] == 11 & ]
PROG
(PARI) is(n)=sum(i=1, n, eulerphi(i)==n)==11 \\ Charles R Greathouse IV, Mar 03 2014
CROSSREFS
Cf. A007366, A007367.
KEYWORD
nonn
AUTHOR
Robert G. Wilson v, Apr 18 2001
STATUS
approved
A293928 Totients phi(m) having one or more solutions to phi(m)^(k+1) = phi(phi(m)^k*m), k >= 1, m >= 1. +10
2
1, 2, 4, 6, 8, 12, 16, 18, 20, 24, 32, 36, 40, 48, 54, 64, 72, 80, 84, 96, 100, 108, 120, 128, 144, 160, 162, 168, 192, 200, 216, 240, 252, 256, 272, 288, 312, 320, 324, 336, 360, 384, 400, 432, 440, 480, 486, 500, 504, 512, 544, 576, 588, 600, 624, 640, 648, 672, 684 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
"Totients" are terms of A000010. - N. J. A. Sloane, Oct 22 2017
The smallest totient absent from the list is 10. This is because the totient inverses of 10, 11 and 22 are not solutions of phi(m)^(k+1) = phi(phi(m)^k*m), k >= 1, m >= 1.
The formula is recursive. For example, taking a(22) we get the following: 11664 = phi(108*324), 1259712 = phi(11664*324), 136048896 = phi(1259712*324), ...
If a solution exists then the smallest value of k must be 1. This follows from a|b implies phi(a)|phi(b), and for k >= 1 a^(k-1)|a^k.
Where (if ever) does this first differ from A068997? - R. J. Mathar, Oct 30 2017
Apparently the set of the m is A151999. - R. J. Mathar, Mar 25 2024
LINKS
Table of n, a(n) for n=1..59.
Max Alekseyev, PARI scripts for various problems
FORMULA
0 < phi(m)^(k+1) = phi(phi(m)^k*m), k >= 1, m >= 1.
EXAMPLE
96 is a term since 96^2 = phi(96*288), with k=1 and m=288 where phi(288) = 96.
PROG
(PARI) isok(n) = {my(iv = invphi(n)); if (#iv, for (m = 1, #iv, if (n^2 == eulerphi(n*iv[m]), return (1)); ); ); return (0); } \\ using the invphi script by Max Alekseyev; Michel Marcus, Nov 01 2017
CROSSREFS
Cf. A000010, A006511, A032447, A007366.
Subsequence of A002202.
KEYWORD
nonn
AUTHOR
Torlach Rush, Oct 19 2017
EXTENSIONS
More terms from Michel Marcus, Oct 24 2017
STATUS
approved
A297475 Numbers n such that phi(x) = n for more than one value of x, and the smallest such x divides the largest. +10
2
1, 2, 8, 10, 22, 28, 30, 44, 46, 52, 54, 56, 58, 66, 70, 78, 82, 92, 102, 104, 106, 110, 116, 126, 128, 130, 136, 138, 140, 148, 150, 164, 166, 172, 178, 190, 196, 198, 204, 210, 212, 222, 226, 228, 238, 250, 260, 262, 268, 270, 282, 292, 294, 296, 306, 310, 316, 330, 332, 342, 344, 346, 356, 358, 366, 368, 372 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
The larger endpoint is always twice the value of the smaller endpoint.
Conjecture 1: The number of solutions, excluding endpoints is always 0, or an odd number. (known to n = 2 * 10^5)
Conjecture 2: If both endpoints are divisible by 5, then the number of solutions (excluding terms of A007366) is of the form 4k + 1. (known to n = 2 * 10^5)
A007366 is contained in this sequence and the number of solutions, excluding endpoints is always 0.
Terms of this sequence are totients with a single odd totient inverse.
LINKS
Michel Marcus, Table of n, a(n) for n = 1..2441
Max Alekseyev, PARI scripts for various problems
FORMULA
2 = max({phi^-1(n)}) / min({phi^-1(n)}).
0 = A006511(n) mod A002181(n).
EXAMPLE
2 is in the sequence because {phi^-1(2)} = {3,4,6}, and 2 = 6 / 3.
8 is in the sequence because {phi^-1(8)} = {15,...,30}, and 2 = 30 / 15.
10 is in the sequence because {phi^-1(10)} = {11,22}, and 2 = 22 / 11.
MATHEMATICA
With[{nn = 67}, Take[#, nn] &@ Keys@ Select[KeySort@ PositionIndex@ Array[EulerPhi, nn^2], IntegerQ[#2/#1] & @@ {First@ #, Last@ #} &]] (* Michael De Vlieger, Dec 31 2017 *)
PROG
(PARI) isok(n) = my(vx = invphi(n)); (#vx > 1) && ((vecmax(vx) % vecmin(vx)) == 0); \\ Michel Marcus, Jul 18 2018
CROSSREFS
Cf. A002181, A002202, A006511, A007366, A032447.
KEYWORD
nonn
AUTHOR
Torlach Rush, Dec 30 2017
STATUS
approved
A060669 Numbers n such that phi(x) = n has exactly 6 solutions. +10
1
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
LINKS
T. D. Noe, Table of n, a(n) for n = 1..1000
EXAMPLE
12 = phi(13) = phi(21) = phi(26) = phi(28) = phi(36) = phi(42).
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 *)
PROG
(PARI) is(n)=sum(i=1, n, eulerphi(i)==n)==6 \\ Charles R Greathouse IV, Mar 03 2014
CROSSREFS
Cf. A007366, A007367.
KEYWORD
nonn
AUTHOR
Robert G. Wilson v, Apr 18 2001
STATUS
approved
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Last modified August 29 15:03 EDT 2024. Contains 375517 sequences. (Running on oeis4.)