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

1,2

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.

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.

Example

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}

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]

Crossrefs

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

Superset of A002110.

Sequence in context: A121711 A249681 A256561 * A383446 A125597 A366170

Adjacent sequences: A384001 A384002 A384003 * A384005 A384006 A384007

Keyword

nonn

Author

Michael De Vlieger, Jun 10 2025

Status

approved