A007374 -id:A007374 - cp's OEIS frontend

A362186 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.

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

Offset: 0

Amiram Eldar, Apr 10 2023

nonn

Is there any n for which a(n) = -1?

Amiram Eldar, Table of n, a(n) for n = 0..1292

The unitary version of A063507.
Cf. A323410, A362181.
Similar sequences: A007374, A361970.

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]]]

A362181(a(n)) = n.

A066420 Least m such that card(invphi(phi(m)))=n.

Original entry on oeis.org

1, 3, 5, 15, 13, 51, 37, 41, 35, 65, 187, 397, 2269, 1059, 313, 73, 337, 247, 937, 185, 689, 1139, 2057, 403, 2827, 485, 323, 1321, 3697, 241, 769, 9001, 433, 7129, 4201, 527, 577, 1297, 1201, 15937, 3313, 3281, 3379, 949, 3121, 7519, 3889, 779, 1763

Offset: 2

Vladeta Jovovic, Dec 25 2001

nonn

T. D. Noe, Table of n, a(n) for n = 2..1023

Cf. A049283, A007374.

Cf. A224531, A224532.

a(n) = A049283(A007374(n)). - Max Alekseyev, Apr 12 2005

More terms from Max Alekseyev, Apr 12 2005

A130669 Smallest k such that phi(x) = k has exactly n odd solutions.

Original entry on oeis.org

1, 6, 24, 60, 144, 72, 216, 480, 600, 240, 432, 1152, 1296, 1080, 4608, 2016, 720, 2520, 2400, 1440, 5184, 4032, 2880, 5280, 7776, 2160, 5760, 21840, 7560, 9600, 6720, 16560, 5040, 15552, 6480, 13440, 10800, 11520, 20736, 4320, 18144, 12096, 28512, 16800

Offset: 1

Franz Vrabec, Jun 27 2007

nonn

			a(3) = 24 because there are 3 odd solutions (35, 39, 45) to phi(x) = 24 and for every k < 24 the number of odd solutions to phi(x) = k is unequal to 3.

Vladeta Jovovic and Donovan Johnson, Table of n, a(n) for n = 1..1000 (first 100 terms from Vladeta Jovovic)

Cf. A007374, A130670.

A378507 The smallest number k such that the equation phi(phi(x)) = k has exactly n solutions.

Original entry on oeis.org

10, 56, 6, 1, 84, 312, 2, 200, 464, 36, 108, 4, 12, 88, 816, 264, 440, 360, 552, 120, 224, 8, 3696, 1320, 928, 176, 624, 1472, 832, 5728, 24, 4560, 1080, 2000, 16, 2848, 72, 1312, 1872, 80, 1120, 216, 880, 336, 23360, 448, 3808, 10608, 648, 528, 352, 9280, 32

Offset: 2

Amiram Eldar, Nov 29 2024

nonn

The smallest number k such that A378506(k) = n.

If phi(phi(x)) = k has a solution, then according to Carmichael's totient function conjecture there is at least one another number y != x such that phi(y) = phi(x) and then y is also a solution. Therefore, according to this conjecture, a(1) does not exist.

Amiram Eldar, Table of n, a(n) for n = 2..1000
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).
David M. Bressoud, A Course in Computational Number Theory (web page), CNT.m, Computational Number Theory Mathematica package.
Eric Weisstein's World of Mathematics, Carmichael's Totient Function Conjecture.
Wikipedia, Carmichael's totient function conjecture.

Cf. A000010, A010554, A007374, A378506.

s[n_] := Sum[PhiMultiplicity[k], {k, PhiInverse[n]}]; seq[len_] := Module[{v = Table[0, {len+1}], c = 0, k = 1, ns}, While[c < len, ns = s[k]; If[0 < ns <= len + 1 && v[[ns]] == 0, v[[ns]] = k; c++]; k++]; Rest[v]]; seq[30] (* using David M. Bressoud's CNT.m *)

s(n) = vecsum(apply(x -> invphiNum(x), invphi(n))); \\ using Max Alekseyev's invphi.gp
lista(len) = {my(v = vector(len+1), c = 0, k = 1, ns); while(c < len, ns = s(k); if(ns > 0 && ns <= len + 1 && v[ns] == 0, c++; v[ns] = k); k++); vecextract(v,"^1");}

A378510 The least totient number k with exactly n solutions to the equation phi(x) = k, where all the solutions are nontotient numbers (A007617).

Original entry on oeis.org

30, 116, 42, 456, 780, 1140, 1368, 1380, 3420, 4356, 5104, 20196, 9396, 1980, 15876, 8316, 4860, 16380, 79464, 239976, 15720, 69300, 129960, 70000, 90360, 141680, 263160, 835380, 802296, 706680, 236808, 39960, 205800, 2898840, 3200904, 598920, 664440, 2723400

Offset: 2

Amiram Eldar, Nov 29 2024

nonn

The least term k of A378509 such that A014197(k) = n.

Amiram Eldar, Table of n, a(n) for n = 2..70
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).
David M. Bressoud, A Course in Computational Number Theory (web page), CNT.m, Computational Number Theory Mathematica package.

Cf. A000010, A002202, A007374, A007617, A014197, A378509.

seq[len_] := Module[{v = Table[0, {len+1}], c = 0, k = 2, s, ns}, While[c < len, s = PhiInverse[k]; ns = Length[s]; If[0 < ns <= len + 1 && AllTrue[s, PhiMultiplicity[#] == 0 &] && v[[ns]] == 0, v[[ns]] = k; c++]; k += 2]; Rest[v]]; seq[10] (* using David M. Bressoud's CNT.m *)

lista(len) = {my(v = vector(len+1), c = 0, k = 2, s, ns, ans); while(c < len, s = invphi(k); ns = #s; ans = 1; for(i = 1, ns, if(istotient(s[i]), ans = 0; break)); if(ans && ns > 0 && ns <= len + 1 && v[ns] == 0, c++; v[ns] = k); k += 2); vecextract(v,"^1");} \\ using Max Alekseyev's invphi.gp

a(n) >= A007374(n).

A085758 Least m such that phi(x)=2m has exactly n solutions.

Original entry on oeis.org

5, 1, 2, 4, 6, 16, 18, 20, 12, 24, 80, 198, 1134, 352, 156, 36, 168, 108, 468, 72, 312, 528, 880, 180, 1280, 192, 144, 660, 1848, 120, 384, 4500, 216, 3564, 2100, 240, 288, 648, 600, 7968, 1656, 1536, 1620, 432, 1560, 3672, 1944, 360, 840, 2496, 8820, 1008, 576

Offset: 2

Lekraj Beedassy, Jul 22 2003

nonn

Sequence refers to the rank of the first occurrence of n in A032446.

			a(0) = 7. Carmichael conjectured that a(1) doesn't exist.

Cf. A032446.

a(n) = A007374(n)/2 for n > 2. - David Wasserman, Feb 09 2005

More terms from David Wasserman, Feb 09 2005

A300911 a(n) is the smallest positive integer k such that psi(x) = k has exactly n solutions, where psi(x) = A001615(x).

Original entry on oeis.org

2, 1, 6, 84, 12, 24, 48, 168, 96, 72, 192, 144, 384, 672, 360, 960, 432, 288, 3648, 1080, 3600, 720, 2736, 1008, 576, 864, 11664, 6720, 7680, 1152, 7920, 6336, 2016, 1440, 3024, 1728, 2160, 14256, 5040, 2592, 2304, 13440, 9072, 10800, 43008, 26208, 24480, 4608, 2880, 10944, 6480

Offset: 0

Altug Alkan, Mar 15 2018

nonn

			a(3) = 84 because psi(2^2*13) = psi(5*13) = psi(83) = 84 and 84 is the least number with this property.

Cf. A001615, A007368, A007374.

psi[n_] := If[n == 1, 1, n Times @@ (1 + 1/First /@ FactorInteger@n)]; t = Sort[Reverse /@ Tally[Array[psi, 50000]]]; L = {2}; Do[ If[t[[j, 1]] == Length@ L, AppendTo[L, t[[j, 2]]]], {j, Length@t}]; L (* Giovanni Resta, Mar 16 2018 *)

{my(N=5*10^4, c=vectorsmall(N), i); A300911=List(); for(n=1,N,(i=A001615(n))>N||c[i]++); for(n=1,oo, for(i=1,N, c[i]==n && listput(A300911,i) && next(2)); break)} \\ M. F. Hasler, Mar 18 2018

A362489 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.

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, Apr 22 2023

nonn

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?

Amiram Eldar, Table of n, a(n) for n = 0..300

Cf. A091732, A362484, A362485.

Similar sequences: A007374, A063507, A361970, A362186.

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 *)

cp's OEIS Frontend

A362186 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.

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A066420 Least m such that card(invphi(phi(m)))=n.

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A130669 Smallest k such that phi(x) = k has exactly n odd solutions.

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A378507 The smallest number k such that the equation phi(phi(x)) = k has exactly n solutions.

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A378510 The least totient number k with exactly n solutions to the equation phi(x) = k, where all the solutions are nontotient numbers (A007617).

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A085758 Least m such that phi(x)=2m has exactly n solutions.

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A300911 a(n) is the smallest positive integer k such that psi(x) = k has exactly n solutions, where psi(x) = A001615(x).

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A362489 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.

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