A378192 a(1) = 1. For n > 1, a(n) is the number of terms a(i); 1 <= i <= n-1 such that phi(a(i)) = phi(a(n-1)), where phi is Euler's totient function A000010.

1, 1, 2, 3, 1, 4, 2, 5, 1, 6, 3, 4, 5, 2, 7, 1, 8, 3, 6, 7, 2, 9, 3, 8, 4, 9, 4, 10, 5, 6, 11, 1, 10, 7, 5, 8, 9, 6, 12, 10, 11, 2, 11, 3, 13, 1, 12, 12, 13, 2, 13, 3, 14, 7, 8, 14, 9, 10, 15, 1, 14, 11, 4, 15, 2, 15, 3, 16, 4, 17, 1, 16, 5, 16, 6, 18, 12, 17

Offset

1,3

Comments

Every k >= 1 occurs infinitely many times, primes appear in order, and every odd prime a(i) = p is followed by 1, (indicating phi(a(i)) = phi(p) = p-1, a new record in the underlying sequence of totients). The subsequence of records is the natural numbers A000027.

Links

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^16.

Example

a(1) = 1 (given), and phi(1) = 1 implies a(2) = 1. Now there are two terms having phi = 1, so a(3) = 2. Phi(2) = 1 means a(4) = 3, and since phi(3) = 2, the first of its kind so far, a(5) = 1, and so on.
The data can be arranged as an irregular table in which each row starts with a record term, beginning as follows:
 1,1;
 2;
 3,1;
 4,2;
 5,1;
 6,3,4,5,2;
 7,1;
 8,3,6,7,2;
 9,3,8,4,9,4;
 10,5,6;
 11,1,10,7,5,8,9,6;
 12,10,11,2,11,3;
 13,1,12,12,13,2,13,3;
 14,7,8,14,9,10;
 15,1,...
 ....

Mathematica

nn = 120; c[_] := 1; a[1] = j = 1; Do[k = c[EulerPhi[j]]++; Set[{a[n], j}, {k, k}], {n, 2, nn}]; Array[a, nn] (* Michael De Vlieger, Nov 19 2024 *)

Crossrefs

Cf. A000010, A000040, A000027, A006093.

Sequence in context: A083906 A361136 A160541 * A286001 A304106 A022446

Adjacent sequences: A378188 A378189 A378191 * A378197 A378199 A378202

Keyword

nonn,new

Author

David James Sycamore, Nov 19 2024

Status

approved