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

%I A384004 #15 Jun 18 2025 00:31:33
%S A384004 1,2,4,8,6,10,22,12,44,18,24,50,98,36,48,54,224,30,42,70,108,66,78,
%T A384004 162,102,60,138,84,174,260,132,90,126,228,354,120,234,168,350,306,150,
%U A384004 516,408,180,252,552,696,294,240,336,612,378,270,1416,300,702,1332,360
%N A384004 a(n) = smallest k such that A010846(k) = n.
%C A384004 For n > 2, a(n) is composite, since A010846(p) = 2 for prime p.
%C A384004 For n <= 3, a(n) = 2^n; for n > 3, a(n) < 2^n, and a(n) is in A024619.
%C A384004 Smallest k with omega(k) = i is A002110(i).
%C A384004 Conjecture: there are only 8 powerful terms (i.e., in A001694) in the sequence.
%H A384004 Michael De Vlieger, <a href="/A384004/b384004.txt">Table of n, a(n) for n = 1..4647</a>
%H A384004 Michael De Vlieger, <a href="/A384004/a384004.png">Log log scatterplot of a(n)</a>, 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.
%e A384004 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.
%e A384004  n  a(n)  row n of A162306
%e A384004 ----------------------------------------------------------
%e A384004  1:   1   {1}
%e A384004  2:   2   {1, p}
%e A384004  3:   4   {1, p, p^2}
%e A384004  4:   8   {1, p, p^2, p^3}
%e A384004  5:   6   {1, p, q, p^2, p*q}
%e A384004  6:  10   {1, p, p^2, q, p^3, p*q}
%e A384004  7:  22   {1, p, p^2, p^3, q, p^4, p*q}
%e A384004  8:  12   {1, p, q, p^2, p*q, p^3, q^2, p^2*q}
%e A384004  9:  44   {1, p, p^2, p^3, q, p^4, p*q, p^5, p^2*q}
%e A384004 10:  18   {1, p, q, p^2, p*q, p^3, q^2, p^2*q, p^4, p*q^2}
%t A384004 (* First, load the theta program from the algorithms linked in A369609, then: *)
%t A384004 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]
%Y A384004 Cf. A000079, A001222, A010846, A024619, A162306, A244052.
%Y A384004 Superset of A002110.
%K A384004 nonn
%O A384004 1,2
%A A384004 _Michael De Vlieger_, Jun 10 2025