A378506 -id:A378506 - cp's OEIS frontend

A378508 Values taken by phi(phi(m)) (A010554).

1, 2, 4, 6, 8, 10, 12, 16, 18, 20, 22, 24, 28, 32, 36, 40, 44, 48, 52, 54, 56, 60, 64, 72, 80, 82, 84, 88, 92, 96, 100, 104, 108, 112, 120, 128, 130, 132, 144, 156, 160, 162, 164, 168, 172, 176, 178, 180, 184, 190, 192, 200, 204, 208, 212, 216, 220, 224, 232, 238, 240, 250, 252, 256, 260, 264, 272, 276, 280, 288, 292, 300, 312, 320, 324, 328, 336, 344, 348, 352, 356, 358, 360, 368, 380, 384, 396, 400

Offset: 1

Amiram Eldar, Nov 29 2024

nonn

Numbers k such that A378506(k) > 0.

Amiram Eldar, Table of n, a(n) for n = 1..10000
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, A010554, A378506.

Subsequence of A002202.

q[k_] := AnyTrue[PhiInverse[k], PhiMultiplicity[#] > 0 &]; Select[Range[1000], q] (* using David M. Bressoud's CNT.m *)

is(k) = {my(v = invphi(k)); for(i = 1, #v, if(istotient(v[i]), return(1))); 0;} \\ using Max Alekseyev's invphi.gp

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");}

A380594 a(n) is the number of positive integers having 2*n primitive roots.

Original entry on oeis.org

6, 4, 4, 6, 2, 8, 0, 4, 2, 2, 2, 8, 0, 2, 0, 4, 0, 4, 0, 12, 0, 2, 0, 12, 0, 2, 4, 0, 0, 2, 0, 6, 0, 0, 0, 10, 0, 0, 0, 2, 2, 6, 0, 4, 0, 2, 0, 12, 0, 2, 0, 0, 0, 4, 0, 6, 0, 0, 0, 10, 0, 0, 0, 6, 2, 2, 0, 0, 0, 0, 0, 16, 0, 0, 0, 0, 0, 2, 0, 8, 4, 2, 0, 6, 0

Offset: 1

David James Sycamore, Michael De Vlieger, and Amiram Eldar, Jan 27 2025

nonn

Let [n] be the set {k; A046144(k) = 2*n}; n >= 1, then a(n) = |[n]|.

If 2*n is a term in A378508, [n] is nonempty and a(n) > 0. Otherwise, if 2*n is not in A378508 then there is no number having 2*n primitive roots, so a(n) = 0; see Example, and A380604.

			For n = 1, 2*n = 2 and there are 6 distinct numbers having 2 primitive roots; [2] = {5,7,9,10,14,18}; so a(10) = 6.
For n = 5, 2*n = 10 and there are just 2 distinct numbers having 10 primitive roots; [5] = {23,46}; so a(5) = 2.
For n = 7, 2*n = 14 and there are no numbers having 14 primitive roots, so a(7) = 0.
The sets [n] listed in rows start as follows; length of row n = a(n):
  n          [n]                   a(n)
  1    {5,7,9,10,14,18}             6;
  2    {11,13,22,26}                4;
  3    {29,27,30,54}                4;
  4    {17,25,31,34,50,62}          6;
  5    {23,46}                      2;
  6    {29,37,43,49,58,74,86,98}    8;
  7    { }                          0;
  8    {41,61,82,122}               4;
  9    {81,162}                     2;
  10   {67,134}                     2;
  ...

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. A046144, A007617, A010554, A033949, A231772, A378506, A378508, A380604.

a[n_] := Count[Join @@ PhiInverse[PhiInverse[2*n]], ?(IntegerQ[PrimitiveRoot[#]] &)]; Array[a, 100] (* _Amiram Eldar, Jan 27 2025, using David M. Bressoud's CNT.m *)

isA033948(n) = {my(f = factor(n)); lcm(znstar(f)[2]) == eulerphi(f);}
a(n) = {my(v = invphi(2*n), w, c = 0); for(i = 1, #v, c += vecsum(apply(x -> isA033948(x), invphi(v[i])))); c;} \\ Amiram Eldar, Jan 27 2025, using Max Alekseyev's invphi.gp

a(n) <= A378506(2*n), with equality iff n is in A007617.

cp's OEIS Frontend

A378508 Values taken by phi(phi(m)) (A010554).

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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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A380594 a(n) is the number of positive integers having 2*n primitive roots.

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