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A135832 Irregular triangle of Section I primes. Row n contains primes p with 2^n < p < 2^(n+1) and phi^n(p) = 2, where phi^n means n iterations of Euler's totient function.

%I A135832 #12 Jul 03 2024 08:34:56
%S A135832 3,5,7,11,13,17,23,29,31,41,47,53,59,61,83,89,97,101,103,107,113,137,
%T A135832 167,179,193,227,233,239,241,251,257,353,359,389,401,409,443,449,461,
%U A135832 467,479,499,503,641,719,769,773,809,821,823,857,881,887,929,941,953
%N A135832 Irregular triangle of Section I primes. Row n contains primes p with 2^n < p < 2^(n+1) and phi^n(p) = 2, where phi^n means n iterations of Euler's totient function.
%C A135832 Sequence A135833 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. The primes in section I are fairly sparse. All other primes belong to section II. Section III consists only of even numbers. See A058812 for the numbers x for each n.
%H A135832 T. D. Noe, <a href="/A135832/b135832.txt">Rows n = 1..22 of triangle, flattened</a>
%H A135832 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 A135832 Table begins:
%e A135832    3;
%e A135832    5,  7;
%e A135832   11, 13;
%e A135832   17, 23, 29, 31;
%e A135832   41, 47, 53, 59, 61;
%e A135832   83, ...
%t A135832 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 && PrimeQ[ # ]&]], {n,nMax}]; t
%Y A135832 Cf. A135834 (Section II primes).
%K A135832 nonn,tabf
%O A135832 1,1
%A A135832 _T. D. Noe_, Nov 30 2007