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A005239 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.

%I A005239 M2409 #36 Oct 27 2023 10:02:08
%S A005239 3,5,7,11,13,15,17,23,25,29,31,41,47,51,53,55,59,61,83,85,89,97,101,
%T A005239 103,107,113,115,119,121,123,125,137,167,179,187,193,205,221,227,233,
%U A005239 235,239,241,249,251,253,255,257,289,353,359,389,391,401,409
%N A005239 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 A005239 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 A005239 R. K. Guy, Unsolved Problems in Number Theory, B41.
%D A005239 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A005239 T. D. Noe, <a href="/A005239/b005239.txt">Rows n = 1..22 of triangle, flattened</a>
%H A005239 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 A005239 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 A005239 Triangle begins:
%e A005239    3;
%e A005239    5,  7;
%e A005239   11, 13, 15;
%e A005239   17, 23, 25, 29,  31;
%e A005239   41, 47, 51, 53,  55,  59,  61;
%e A005239   83, 85, 89, 97, 101, 103, 107, 113, 115, 119, 121, 123, 125;
%e A005239   ...
%t A005239 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 A005239 Cf. A000010, A058812, A092878.
%Y A005239 Cf. A135832 (Section I primes).
%K A005239 nonn,tabf
%O A005239 1,1
%A A005239 _N. J. A. Sloane_
%E A005239 More terms from _Jud McCranie_, Feb 15 1997
%E A005239 Corrected and extended by _T. D. Noe_, Dec 05 2007