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Irregular table read by rows in which the n-th row consists of all the numbers m such that uphi(m) = n, where uphi is the unitary totient function ( A047994).
+10
15
1, 2, 3, 6, 4, 5, 10, 7, 12, 14, 8, 9, 15, 18, 30, 11, 22, 13, 20, 21, 26, 42, 24, 16, 17, 34, 19, 28, 38, 33, 66, 23, 46, 25, 35, 36, 39, 50, 60, 70, 78, 27, 54, 29, 40, 58, 31, 44, 48, 62, 32, 45, 51, 90, 102, 37, 52, 57, 74, 84, 114, 41, 55, 82, 110, 43, 56, 86
EXAMPLE
The table begins:
n n-th row
-- --------
1 1, 2;
2 3, 6;
3 4;
4 5, 10;
5
6 7, 12, 14;
7 8;
8 9, 15, 18, 30;
9
10 11, 22;
11
12 13, 20, 21, 26, 42;
MATHEMATICA
invUPhi[n_] := Module[{fct = f[n], sol}, sol = Times @@@ (1 + Select[fct, UnsameQ @@ # && (Length[#] == 1 || CoprimeQ @@ (# + 1)) && Times @@ PrimeNu[# + 1] == 1 &]); Sort@ Join[sol, 2*Select[sol, OddQ]]]; invUPhi[1] = {1, 2}; Table[invUPhi[n], {n, 1, 50}] // Flatten (* using the function f by T. D. Noe at A162247 *)
Number of numbers k such that uphi(k) = n, where uphi is the unitary totient function ( A047994).
+10
9
2, 2, 1, 2, 0, 3, 1, 4, 0, 2, 0, 5, 0, 1, 1, 2, 0, 3, 0, 2, 0, 2, 0, 8, 0, 2, 0, 3, 0, 4, 1, 4, 0, 0, 0, 6, 0, 0, 0, 4, 0, 3, 0, 2, 0, 2, 0, 11, 0, 0, 0, 2, 0, 1, 0, 4, 0, 2, 0, 8, 0, 1, 1, 2, 0, 3, 0, 0, 0, 3, 0, 11, 0, 0, 0, 0, 0, 3, 0, 8, 0, 2, 0, 5, 0, 0, 0
MATHEMATICA
a[n_] := Length[invUPhi[n]]; Array[a, 100] (* using the function invUPhi from A361966 *)
Unitary highly totient numbers: numbers k that have more solutions x to the equation uphi(x) = k than any smaller k, where uphi is the unitary totient function ( A047994).
+10
8
1, 6, 8, 12, 24, 48, 96, 120, 144, 240, 480, 576, 720, 1440, 2880, 4320, 5760, 8640, 10080, 17280, 20160, 30240, 34560, 40320, 60480, 80640, 120960, 241920, 362880, 483840, 725760, 967680, 1209600, 1451520, 2177280, 2419200, 2903040, 3628800, 4354560, 4838400
COMMENTS
The corresponding numbers of solutions are 2, 3, 4, 5, 8, 11, ... ( A361971).
MATHEMATICA
solnum[n_] := Length[invUPhi[n]]; seq[kmax_] := Module[{s = {}, solmax=0}, Do[sol = solnum[k]; If[sol > solmax, solmax = sol; AppendTo[s, k]], {k, 1, kmax}]; s]; seq[10^5] (* using the function invUPhi from A361966 *)
Numbers k with a single solution x to the equation uphi(x) = k, where uphi is the unitary totient function ( A047994).
+10
8
3, 7, 14, 15, 31, 54, 62, 63, 127, 154, 174, 182, 186, 234, 246, 254, 255, 294, 308, 318, 322, 364, 406, 414, 496, 510, 511, 516, 534, 558, 574, 594, 644, 666, 678, 762, 804, 806, 812, 846, 870, 948, 1022, 1023, 1026, 1036, 1074, 1098, 1146, 1148, 1164, 1204, 1246
COMMENTS
Numbers k such that A361967(k) = 1. According to Carmichael's totient function conjecture, there are no numbers with a single solution x to the corresponding equation phi(x) = k, with Euler's totient function ( A000010). A000225(m) = 2^m - 1 is a term for all m >= 2. These are the only odd terms.
MATHEMATICA
Select[Range[1250], Length[invUPhi[#]] == 1 &] (* using the function invUPhi from A361966 *)
a(n) is the least number k such that the equation A323410(x) = k has exactly n solutions, or -1 if no such k exists.
+10
3
2, 0, 6, 10, 20, 31, 47, 53, 65, 77, 89, 113, 125, 119, 149, 173, 167, 179, 233, 279, 239, 209, 439, 293, 365, 299, 329, 359, 455, 521, 467, 389, 461, 419, 479, 773, 539, 509, 599, 845, 671, 791, 749, 719, 659, 629, 809, 1055, 881, 779, 899, 965, 929, 1121, 839, 1403
COMMENTS
Is there any n for which a(n) = -1?
MATHEMATICA
ucototient[n_] := n - Times @@ (Power @@@ FactorInteger[n] - 1); ucototient[1] = 0; With[{max = 300}, solnum = Table[0, {n, 1, max}]; Do[If[(i = ucototient[k]) <= max, solnum[[i]]++], {k, 2, max^2}]; Join[{2, 0}, TakeWhile[FirstPosition[ solnum, #] & /@ Range[2, max] // Flatten, NumberQ]]]
a(n) is the least number k such that the equation iphi(x) = k has exactly 2*n solutions, or -1 if no such k exists, where iphi is the infinitary totient function A091732.
+10
1
5, 1, 6, 12, 36, 24, 396, 48, 216, 96, 528, 144, 384, 2784, 432, 240, 1296, 288, 1584, 1800, 480, 1680, 1080, 864, 576, 3240, 2016, 960, 6624, 720, 1152, 7776, 12000, 8448, 5280, 1728, 10752, 2304, 4032, 4800, 6048, 3840, 2160, 5184, 4608, 6336, 1440, 10560, 29568
COMMENTS
a(n) is the least number k such that A362485(k) = 2*n. Odd values of A362485 are impossible. Is there any n for which a(n) = -1?
MATHEMATICA
solnum[n_] := Length[invIPhi[n]]; seq[len_, kmax_] := Module[{s = Table[-1, {len}], c = 0, k = 1, ind}, While[k < kmax && c < len, ind = solnum[k]/2 + 1; If[ind <= len && s[[ind]] < 0, c++; s[[ind]] = k]; k++]; s]; seq[50, 10^5] (* using the function invIPhi from A362484 *)
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