%I M2409 #36 Oct 27 2023 10:02:08

%S 3,5,7,11,13,15,17,23,25,29,31,41,47,51,53,55,59,61,83,85,89,97,101,

%T 103,107,113,115,119,121,123,125,137,167,179,187,193,205,221,227,233,

%U 235,239,241,249,251,253,255,257,289,353,359,389,391,401,409

%N Irregular triangle of Section I numbers. Row n contains numbers k with 2^n < k < 2^(n+1) and phi^n(k) = 2, where phi^n means n iterations of Euler's totient function.

%C Sequence A092878 gives the number of terms in row n. Shapiro describes how the numbers x with phi^n(x)=2 can be divided into 3 sections: I: 2^n < x < 2^(n+1), II: 2^(n+1) <= x <= 3^n and III: 3^n < x <= 2*3^n. See A058812 for the numbers x for each n. - _T. D. Noe_, Dec 05 2007

%D R. K. Guy, Unsolved Problems in Number Theory, B41.

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H T. D. Noe, <a href="/A005239/b005239.txt">Rows n = 1..22 of triangle, flattened</a>

%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 Harold Shapiro, <a href="http://www.jstor.org/stable/2303988">An arithmetic function arising from the phi function</a>, Amer. Math. Monthly, Vol. 50, No. 1 (1943), 18-30.

%e Triangle begins:

%e 3;

%e 5, 7;

%e 11, 13, 15;

%e 17, 23, 25, 29, 31;

%e 41, 47, 51, 53, 55, 59, 61;

%e 83, 85, 89, 97, 101, 103, 107, 113, 115, 119, 121, 123, 125;

%e ...

%t nMax=10; nn=2^nMax; c=Table[0,{nn}]; Do[c[[n]]=1+c[[EulerPhi[n]]], {n,2,nn}]; t={}; Do[t=Join[t,Select[Flatten[Position[c,n]], #<2^n&]], {n,nMax}]; t (* _T. D. Noe_, Dec 05 2007 *)

%Y Cf. A000010, A058812, A092878.

%Y Cf. A135832 (Section I primes).

%K nonn,tabf

%O 1,1

%A _N. J. A. Sloane_

%E More terms from _Jud McCranie_, Feb 15 1997

%E Corrected and extended by _T. D. Noe_, Dec 05 2007