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.

3, 5, 7, 11, 13, 17, 23, 29, 31, 41, 47, 53, 59, 61, 83, 89, 97, 101, 103, 107, 113, 137, 167, 179, 193, 227, 233, 239, 241, 251, 257, 353, 359, 389, 401, 409, 443, 449, 461, 467, 479, 499, 503, 641, 719, 769, 773, 809, 821, 823, 857, 881, 887, 929, 941, 953

Offset: 1

Author

T. D. Noe, Nov 30 2007

Keywords

• nonn
• tabf

Comments

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.

Examples

			Table begins:
   3;
   5,  7;
  11, 13;
  17, 23, 29, 31;
  41, 47, 53, 59, 61;
  83, ...
		

Links

• T. D. Noe, Rows n = 1..22 of triangle, flattened
• Harold Shapiro, An arithmetic function arising from the phi function, Amer. Math. Monthly, Vol. 50, No. 1 (1943), 18-30.

Crossrefs

Cf. A135834 (Section II primes).

Programs

Mathematica

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