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. (Formerly M2409)

3, 5, 7, 11, 13, 15, 17, 23, 25, 29, 31, 41, 47, 51, 53, 55, 59, 61, 83, 85, 89, 97, 101, 103, 107, 113, 115, 119, 121, 123, 125, 137, 167, 179, 187, 193, 205, 221, 227, 233, 235, 239, 241, 249, 251, 253, 255, 257, 289, 353, 359, 389, 391, 401, 409

Offset

1,1

Comments

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

References

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

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

Links

T. D. Noe, Rows n = 1..22 of triangle, flattened

T. D. Noe, Primes in classes of the iterated totient function, JIS 11 (2008) 08.1.2.

Harold Shapiro, An arithmetic function arising from the phi function, Amer. Math. Monthly, Vol. 50, No. 1 (1943), 18-30.

Example

Triangle begins:
   3;
   5,  7;
  11, 13, 15;
  17, 23, 25, 29,  31;
  41, 47, 51, 53,  55,  59,  61;
  83, 85, 89, 97, 101, 103, 107, 113, 115, 119, 121, 123, 125;
  ...

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&]], {n, nMax}]; t (* T. D. Noe, Dec 05 2007 *)

Crossrefs

Cf. A000010, A058812, A092878.

Cf. A135832 (Section I primes).

Sequence in context: A179458 A062086 A283680 * A141107 A047484 A036991

Adjacent sequences: A005236 A005237 A005238 * A005240 A005241 A005242

Keyword

nonn,tabf

Author

N. J. A. Sloane

Extensions

More terms from Jud McCranie, Feb 15 1997

Corrected and extended by T. D. Noe, Dec 05 2007

Status

approved