

A058812


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


7



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
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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 inversephi 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 nth 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.


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/(p1)); n = m; nn = {}; While[n <= nmax, If[EulerPhi[n] == m, AppendTo[nn, n]]; n++]; nn]; row[n_] := row[n] = inversePhi /@ row[n1] // Flatten // Union; row[0] = {1}; row[1] = {2}; Table[row[n], {n, 0, 5}] // Flatten (* JeanFrançois 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.
Sequence in context: A129607 A130340 A130339 * 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



