A379883 a(1) = 1. Let j = a(n-1) and r = A046144(j). Then for n > 1, if j is novel and r > 0, a(n) = r. If j is novel and r = 0 then a(n) = 1. If j has occurred k (>1) times already then a(n) = k*j.

1, 1, 2, 1, 3, 1, 4, 1, 5, 2, 4, 8, 1, 6, 1, 7, 2, 6, 12, 1, 8, 16, 1, 9, 2, 8, 24, 1, 10, 2, 10, 20, 1, 11, 4, 12, 24, 48, 1, 12, 36, 1, 13, 4, 16, 32, 1, 14, 2, 12, 48, 96, 1, 15, 1, 16, 48, 144, 1, 17, 8, 32, 64, 1, 18, 2, 14, 28, 1, 19, 6, 18, 36, 72, 1, 20, 40, 1, 21, 1, 22, 4, 20, 60, 1, 23, 10, 30, 1, 24, 72, 144, 288, 1, 25, 8, 40, 80, 1, 26, 4

Offset: 1

Author

David James Sycamore, Jan 09 2025

Keywords

• nonn
• easy

Comments

In other words if j = a(n-1) has not occurred earlier and has r (> 0) primitive roots then a(n) = r. Cases where novel A046144(j) = 0 cannot be counted multiplicatively (as k*j) for repeats, so a(n) = 1 is designed to permit the sequence to continue past such points, which means including in the count of 1's terms following (1,2,3,4,6), for which it is true that r = 1. Terms beyond a(12) = 8 which count the number of 1's (by the second condition) give the cardinality of terms with no primitive roots, plus the few (5) cases of terms with primitive root = 1.

Every even number m in A380594 appears finitely many times, consequent to occasions of integers v (>6) for which A046144(v) = m, and to repetitions (k*j) = m for j even. However every odd number appears once only (consequent to odd counts of 1's). The odd numbers appear in order, and since 2 precedes all of them, the primes are in order.

Examples

			a(2) = 1 since a(1)=1 and and 1 has one primitive root. Since 1 has been seen twice, a(3) = 2 and then a(4) = 1 since 2 is a novel term with one primitive root.
a(9) = 5, a novel term with two primitive roots so a(10) = 2, which has appeared once before (a(3)=2), so a(11) = 4, the second occurrence of 4 so a(12) = 8, a novel term with no primitive roots, meaning that a(13) = 1. The count of 1's is now 6, so a(14) = 6, meaning 5 prior terms with one primitive root and one with none.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000
• Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^20.

Crossrefs

Cf. A046144, A378508, A380594, A380604.

Programs

Mathematica

nn = 120; c[_] := 0; j = 1;
f[x_] := f[x] = Which[
  x == 1, 1,
  IntegerQ[PrimitiveRoot[x]], Nest[EulerPhi, x, 2],
  True, 0];
{j}~Join~Reap[Monitor[Do[
  If[c[j] == 0,
    Set[k, # + Boole[# == 0]] &[f[j]]; c[j]++,
    k = ++c[j]*j ];
j = Sow[k], {n, 2, nn}], n] ][[-1, 1]] (* Michael De Vlieger, Jan 09 2025 *)

Extensions

a(78)=1 inserted by David Radcliffe, Aug 03 2025