A384004 a(n) = smallest k such that A010846(k) = n.

1, 2, 4, 8, 6, 10, 22, 12, 44, 18, 24, 50, 98, 36, 48, 54, 224, 30, 42, 70, 108, 66, 78, 162, 102, 60, 138, 84, 174, 260, 132, 90, 126, 228, 354, 120, 234, 168, 350, 306, 150, 516, 408, 180, 252, 552, 696, 294, 240, 336, 612, 378, 270, 1416, 300, 702, 1332, 360

Offset: 1

Author

Michael De Vlieger, Jun 10 2025

Keywords

• nonn

Comments

For n > 2, a(n) is composite, since A010846(p) = 2 for prime p.

For n <= 3, a(n) = 2^n; for n > 3, a(n) < 2^n, and a(n) is in A024619.

Smallest k with omega(k) = i is A002110(i).

Conjecture: there are only 8 powerful terms (i.e., in A001694) in the sequence.

Examples

			Table of n, a(n) for n=1..10, showing row a(n) of A162306, replacing lpf(a(n)) with p, and A119288(a(n)) with q. Note: A010846(n) is the length of row n of A162306.
 n  a(n)  row n of A162306
----------------------------------------------------------
 1:   1   {1}
 2:   2   {1, p}
 3:   4   {1, p, p^2}
 4:   8   {1, p, p^2, p^3}
 5:   6   {1, p, q, p^2, p*q}
 6:  10   {1, p, p^2, q, p^3, p*q}
 7:  22   {1, p, p^2, p^3, q, p^4, p*q}
 8:  12   {1, p, q, p^2, p*q, p^3, q^2, p^2*q}
 9:  44   {1, p, p^2, p^3, q, p^4, p*q, p^5, p^2*q}
10:  18   {1, p, q, p^2, p*q, p^3, q^2, p^2*q, p^4, p*q^2}
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..4647
• Michael De Vlieger, Log log scatterplot of a(n), n = 1..4647, showing primes in large red, proper prime powers in large gold, composite primorials in large bright green, other squarefree composites in small green, and numbers neither squarefree nor prime powers in blue or magenta, with magenta signifying powerful numbers that are not prime powers.

Crossrefs

Cf. A000079, A001222, A010846, A024619, A162306, A244052.

Superset of A002110.

Programs

Mathematica

(* First, load the theta program from the algorithms linked in A369609, then: *)
nn = 2310; t[_] := 0; u = 1; Do[(If[t[#] == 0, t[#] = n]; If[# == u, While[t[u] != 0, u++]]) &[theta[n]], {n, nn}]; Array[t, u - 1]