A378192 - OEIS

%I #19 Nov 20 2024 09:43:57

%S 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,

%T 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,

%U 14,11,4,15,2,15,3,16,4,17,1,16,5,16,6,18,12,17

%N 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.

%C 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.

%H Michael De Vlieger, <a href="/A378192/b378192.txt">Table of n, a(n) for n = 1..10000</a>

%H Michael De Vlieger, <a href="/A378192/a378192.png">Log log scatterplot of a(n)</a>, n = 1..2^16.

%e 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.

%e The data can be arranged as an irregular table in which each row starts with a record term, beginning as follows:

%e  1,1;

%e  2;

%e  3,1;

%e  4,2;

%e  5,1;

%e  6,3,4,5,2;

%e  7,1;

%e  8,3,6,7,2;

%e  9,3,8,4,9,4;

%e  10,5,6;

%e  11,1,10,7,5,8,9,6;

%e  12,10,11,2,11,3;

%e  13,1,12,12,13,2,13,3;

%e  14,7,8,14,9,10;

%e  15,1,...

%e  ....

%t 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 *)

%Y Cf. A000010, A000040, A000027, A006093.

%K nonn,new

%O 1,3

%A _David James Sycamore_, Nov 19 2024