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

Author

Labos Elemer, Jan 03 2001

Keywords

• nonn
• tabf
• nice
• look

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

Examples

			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
		

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

Crossrefs

Cf. A000010, A007617, A005277.

A058811 gives the number of terms in each row.

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

Cf. also A334111.

Programs

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 *)

Extensions

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

New name from David A. Corneth, Mar 26 2019