A092878 -id:A092878 - cp's OEIS frontend

A058812 Irregular triangle of rows of numbers in increasing order. Row 1 = {1}. Row m + 1 contains all numbers k such that phi(k) is in row m.

1, 2, 3, 4, 6, 5, 7, 8, 9, 10, 12, 14, 18, 11, 13, 15, 16, 19, 20, 21, 22, 24, 26, 27, 28, 30, 36, 38, 42, 54, 17, 23, 25, 29, 31, 32, 33, 34, 35, 37, 39, 40, 43, 44, 45, 46, 48, 49, 50, 52, 56, 57, 58, 60, 62, 63, 66, 70, 72, 74, 76, 78, 81, 84, 86, 90, 98, 108, 114, 126

Offset: 0

Labos Elemer, Jan 03 2001

Nontotient values (A007617) are also present as inverses of some previous value.

Old name was: Irregular triangle of inverse totient values of integers generated recursively. Initial value is 1. The inverse-phi sets in increasing order are as follows: {1} -> {2} -> {3, 4, 6} -> {5, 7, 8, 9, 10, 12, 14, 18} -> ... The terms of each row are arranged by magnitude. The next row starts when the increase of terms is violated. 2^n is included in the n-th row. - David A. Corneth, Mar 26 2019

			Triangle begins:
  1;
  2;
  3, 4, 6;
  5, 7, 8, 9, 10, 12, 14, 18;
  ...
Row 3 is {3, 4, 6} as for each number k in this row, phi(k) is in row 2. - _David A. Corneth_, Mar 26 2019

T. D. Noe, Rows n=0..9 of triangle, flattened
Hartosh Singh Bal, Gaurav Bhatnagar, Prime number conjectures from the Shapiro class structure, arXiv:1903.09619 [math.NT], 2019.
T. D. Noe, Primes in classes of the iterated totient function, JIS 11 (2008) 08.1.2.
Index entries for sequences that are permutations of the natural numbers

Cf. A000010, A007617, A005277.
A058811 gives the number of terms in each row.
Cf. A003434, A007755, A060611, A092878, A098196, A135832.
Cf. also A334111.

inversePhi[m_?OddQ] = {}; inversePhi[1] = {1, 2}; inversePhi[m_] := Module[{p, nmax, n, nn}, p = Select[Divisors[m] + 1, PrimeQ]; nmax = m*Times @@ (p/(p-1)); n = m; nn = {}; While[n <= nmax, If[EulerPhi[n] == m, AppendTo[nn, n]]; n++]; nn]; row[n_] := row[n] = inversePhi /@ row[n-1] // Flatten // Union; row[0] = {1}; row[1] = {2}; Table[row[n], {n, 0, 5}] // Flatten (* Jean-François Alcover, Dec 06 2012 *)

Definition revised by T. D. Noe, Nov 30 2007

New name from David A. Corneth, Mar 26 2019

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.

Original entry on oeis.org

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

N. J. A. Sloane

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

			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;
  ...

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).

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.

Cf. A000010, A058812, A092878.

Cf. A135832 (Section I primes).

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 *)

More terms from Jud McCranie, Feb 15 1997

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

A135833 Number of Section I primes between 2^n and 2^(n+1). See A135832.

Original entry on oeis.org

1, 2, 2, 4, 5, 7, 9, 13, 18, 21, 28, 43, 56, 62, 72, 94, 133, 142, 187, 241, 313, 376, 436, 517, 709, 858, 982, 1271, 1561, 1814, 2192, 2658, 3184, 3853, 4601, 5648, 6881, 8009, 9535, 11651, 13712, 16325, 19381, 23323, 27097, 31782, 37924, 44673, 52695, 62147

Offset: 1

T. D. Noe, Nov 30 2007

nonn

Comparing these numbers with A036378, the number of primes between 2^n and 2^(n+1), leads one to conjecture that the density of Section I primes is 0.

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

T. D. Noe, Computing Numbers in Section I of the Totient Iteration

Cf. A092878 (number of odd numbers in Section I).

class[ n_ ] := Length[ NestWhileList[ EulerPhi,n,#>2& ] ]-1; k=2; Table[ cnt=0; While[ p=Prime[ k ]; p<2^(n+1), If[ class[ p ]==n, cnt++ ]; k++ ]; cnt, {n,20} ] (* T. D. Noe, Aug 04 2008 *)

More terms from T. D. Noe, Aug 04 2008

Extension. T. D. Noe, Nov 18 2008

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A058812 Irregular triangle of rows of numbers in increasing order. Row 1 = {1}. Row m + 1 contains all numbers k such that phi(k) is in row m.

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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.

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A135833 Number of Section I primes between 2^n and 2^(n+1). See A135832.

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Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Extensions