Search: a007366 -id:a007366
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A007367
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Numbers k such that phi(x) = k has exactly 3 solutions.
(Formerly M2163)
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+10
14
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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
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OFFSET
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1,1
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COMMENTS
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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
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REFERENCES
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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).
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LINKS
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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].
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EXAMPLE
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phi(69) = phi(92) = phi(138) = 44, so 44 is a term.
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MATHEMATICA
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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 *)
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PROG
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(Haskell)
a007367 n = a007367_list !! (n-1)
a007367_list = map fst $
filter ((== 3) . snd) $ zip a002202_list a058277_list
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A071388
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Numbers k such that the cardinality of the set of solutions to phi(x) = k is a prime.
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+10
4
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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
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OFFSET
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1,2
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COMMENTS
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LINKS
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EXAMPLE
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InvPhi[48]={65,104,105,112,130,140,144,156,168,180,210} has 11 terms, so 48 is here.
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MAPLE
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filter:= n -> isprime(nops(numtheory:-invphi(n))):
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A060667
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Numbers n such that phi(x) = n has exactly 4 solutions.
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+10
2
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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
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OFFSET
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1,1
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LINKS
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EXAMPLE
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18 = phi(19) = phi(27) = phi(38) = phi(54).
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MATHEMATICA
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a = Table[ 0, {2500} ]; Do[ p = EulerPhi[ n ]; If[ p < 2501, a[ [ p ] ]++ ], {n, 1, 5000} ]; Select[ Range[ 2500 ], a[ [ # ] ] == 4 & ]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A060668
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Numbers n such that phi(x) = n has exactly 5 solutions.
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+10
2
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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
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OFFSET
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1,1
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LINKS
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EXAMPLE
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8 = phi(15) = phi(16) = phi(20) = phi(24) = phi(30).
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MATHEMATICA
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a = Table[ 0, {5000} ]; Do[ p = EulerPhi[ n ]; If[ p < 5001, a[[ p ]]++ ], {n, 1, 15000} ]; Select[ Range[ 5000 ], a[[ # ]] == 5 & ]
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PROG
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A060670
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Numbers n such that phi(x) = n has exactly 7 solutions.
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+10
2
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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
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OFFSET
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1,1
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LINKS
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EXAMPLE
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32 = phi(51) = phi(64) = phi(68) = phi(80) = phi(96) = phi(102) = phi(120).
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MATHEMATICA
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a = Table[ 0, {8000} ]; Do[ p = EulerPhi[ n ]; If[ p < 8001, a[ [ p ] ]++ ], {n, 1, 25000} ]; Select[ Range[ 8000 ], a[ [ # ] ] == 7 & ]
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PROG
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A060671
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Numbers n such that phi(x) = n has exactly 8 solutions.
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+10
2
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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
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OFFSET
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1,1
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LINKS
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EXAMPLE
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36 = phi(37) = phi(57) = phi(63) = phi(74) = phi(76) = phi(108) = phi(114) = phi(126).
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MATHEMATICA
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a = Table[ 0, {5000} ]; Do[ p = EulerPhi[ n ]; If[ p < 5001, a[ [ p ] ]++ ], {n, 1, 25000} ]; Select[ Range[ 5000 ], a[ [ # ] ] == 8 & ]
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PROG
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A060674
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Numbers n such that phi(x) = n has exactly 11 solutions.
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+10
2
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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
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OFFSET
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1,1
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LINKS
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EXAMPLE
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48 = phi(65) = phi(104) = phi(105) = phi(112) = phi(130) = phi(140) = phi(144) = phi(156) = phi(168) = phi(180) = phi(210).
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MATHEMATICA
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a = Table[ 0, {12500} ]; Do[ p = EulerPhi[ n ]; If[ p < 12501, a[ [ p ] ]++ ], {n, 1, 50000} ]; Select[ Range[ 12500 ], a[ [ # ] ] == 11 & ]
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PROG
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A293928
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Totients phi(m) having one or more solutions to phi(m)^(k+1) = phi(phi(m)^k*m), k >= 1, m >= 1.
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+10
2
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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
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OFFSET
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1,2
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COMMENTS
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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.
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LINKS
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FORMULA
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0 < phi(m)^(k+1) = phi(phi(m)^k*m), k >= 1, m >= 1.
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EXAMPLE
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96 is a term since 96^2 = phi(96*288), with k=1 and m=288 where phi(288) = 96.
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A297475
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Numbers n such that phi(x) = n for more than one value of x, and the smallest such x divides the largest.
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+10
2
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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
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OFFSET
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1,2
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COMMENTS
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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.
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LINKS
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FORMULA
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2 = max({phi^-1(n)}) / min({phi^-1(n)}).
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EXAMPLE
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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.
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MATHEMATICA
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With[{nn = 67}, Take[#, nn] &@ Keys@ Select[KeySort@ PositionIndex@ Array[EulerPhi, nn^2], IntegerQ[#2/#1] & @@ {First@ #, Last@ #} &]] (* Michael De Vlieger, Dec 31 2017 *)
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PROG
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(PARI) isok(n) = my(vx = invphi(n)); (#vx > 1) && ((vecmax(vx) % vecmin(vx)) == 0); \\ Michel Marcus, Jul 18 2018
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A060669
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Numbers n such that phi(x) = n has exactly 6 solutions.
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+10
1
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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
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OFFSET
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1,1
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LINKS
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EXAMPLE
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12 = phi(13) = phi(21) = phi(26) = phi(28) = phi(36) = phi(42).
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MATHEMATICA
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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 *)
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PROG
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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