%I #35 May 27 2020 10:31:54
%S 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,
%T 36,38,42,54,17,23,25,29,31,32,33,34,35,37,39,40,43,44,45,46,48,49,50,
%U 52,56,57,58,60,62,63,66,70,72,74,76,78,81,84,86,90,98,108,114,126
%N 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.
%C Nontotient values (A007617) are also present as inverses of some previous value.
%C 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
%H T. D. Noe, <a href="/A058812/b058812.txt">Rows n=0..9 of triangle, flattened</a>
%H Hartosh Singh Bal, Gaurav Bhatnagar, <a href="https://arxiv.org/abs/1903.09619">Prime number conjectures from the Shapiro class structure</a>, arXiv:1903.09619 [math.NT], 2019.
%H T. D. Noe, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL11/Noe/noe080107.html">Primes in classes of the iterated totient function</a>, JIS 11 (2008) 08.1.2.
%H <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%e Triangle begins:
%e 1;
%e 2;
%e 3, 4, 6;
%e 5, 7, 8, 9, 10, 12, 14, 18;
%e ...
%e 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
%t 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 *)
%Y Cf. A000010, A007617, A005277.
%Y A058811 gives the number of terms in each row.
%Y Cf. A003434, A007755, A060611, A092878, A098196, A135832.
%Y Cf. also A334111.
%K nonn,tabf,nice,look
%O 0,2
%A _Labos Elemer_, Jan 03 2001
%E Definition revised by _T. D. Noe_, Nov 30 2007
%E New name from _David A. Corneth_, Mar 26 2019