A058812 Irregular triangle of rows of numbers in increasing order. Row 1 = {1}. Row m + 1 contains all numbers k such that phi(k) is in row m.

1, 2, 3, 4, 6, 5, 7, 8, 9, 10, 12, 14, 18, 11, 13, 15, 16, 19, 20, 21, 22, 24, 26, 27, 28, 30, 36, 38, 42, 54, 17, 23, 25, 29, 31, 32, 33, 34, 35, 37, 39, 40, 43, 44, 45, 46, 48, 49, 50, 52, 56, 57, 58, 60, 62, 63, 66, 70, 72, 74, 76, 78, 81, 84, 86, 90, 98, 108, 114, 126

Offset

0,2

Comments

Nontotient values (A007617) are also present as inverses of some previous value.

Old name was: Irregular triangle of inverse totient values of integers generated recursively. Initial value is 1. The inverse-phi sets in increasing order are as follows: {1} -> {2} -> {3, 4, 6} -> {5, 7, 8, 9, 10, 12, 14, 18} -> ... The terms of each row are arranged by magnitude. The next row starts when the increase of terms is violated. 2^n is included in the n-th row. - David A. Corneth, Mar 26 2019

Links

T. D. Noe, Rows n=0..9 of triangle, flattened

Hartosh Singh Bal, Gaurav Bhatnagar, Prime number conjectures from the Shapiro class structure, arXiv:1903.09619 [math.NT], 2019.

T. D. Noe, Primes in classes of the iterated totient function, JIS 11 (2008) 08.1.2.

Index entries for sequences that are permutations of the natural numbers

Example

Triangle begins:
  1;
  2;
  3, 4, 6;
  5, 7, 8, 9, 10, 12, 14, 18;
  ...
Row 3 is {3, 4, 6} as for each number k in this row, phi(k) is in row 2. - David A. Corneth, Mar 26 2019

Mathematica

inversePhi[m_?OddQ] = {}; inversePhi[1] = {1, 2}; inversePhi[m_] := Module[{p, nmax, n, nn}, p = Select[Divisors[m] + 1, PrimeQ]; nmax = m*Times @@ (p/(p-1)); n = m; nn = {}; While[n <= nmax, If[EulerPhi[n] == m, AppendTo[nn, n]]; n++]; nn]; row[n_] := row[n] = inversePhi /@ row[n-1] // Flatten // Union; row[0] = {1}; row[1] = {2}; Table[row[n], {n, 0, 5}] // Flatten (* Jean-François Alcover, Dec 06 2012 *)

Crossrefs

Cf. A000010, A007617, A005277.

A058811 gives the number of terms in each row.

Cf. A003434, A007755, A060611, A092878, A098196, A135832.

Cf. also A334111.

Sequence in context: A130340 A130339 A350348 * A320454 A320455 A072796

Adjacent sequences: A058809 A058810 A058811 * A058813 A058814 A058815

Keyword

nonn,tabf,nice,look

Author

Labos Elemer, Jan 03 2001

Extensions

Definition revised by T. D. Noe, Nov 30 2007

New name from David A. Corneth, Mar 26 2019

Status

approved