A373737 a(n) is the smallest number k in the sorted sequence S(q) = {k : rad(k) = q}, q = A120944(n), such that tau(k) - A008479(k) is not positive, where rad = A007947 and tau = A000005.

162, 250, 686, 1875, 7203, 2662, 4394, 750, 3993, 578, 12005, 722, 6591, 2058, 1058, 14739, 73205, 20577, 1682, 1922, 142805, 5346, 36501, 3430, 2738, 102487, 6318, 3362, 417605, 3698, 73167, 199927, 89373, 4418, 651605, 5202, 25725, 5618, 13310, 151959, 6498

Offset: 1

Author

Michael De Vlieger, Jun 24 2024

Keywords

• nonn
• easy

Comments

Numbers k whose position i in S(n) is such that tau(k) <= i, i.e., that A372720(k) is not positive.

For k = p^m, m > 0, in S(p), p prime, tau(p^m) > A008479(p^m) since tau(p^m) = m + 1 and A008479(p^m) = m. Therefore we consider only composite squarefree q in this sequence.

a(n) is in A126706.

Conjecture: a(n) <= s*gpf(s)^floor(log_gpf(s) s^2), where gpf = A006530.

Examples

			a(1) = 162 since the 12th term in S(6) = A033845 = {6, 12, 18, 24, 36, 48, 54, ..., 162, ...} is the smallest k = S(6, i) such that tau(S(6, i)) <= i: tau(162) = 10 while i = 12.
a(2) = 250 since S(10, 9) = 250 gives tau(250) = 8, and 8 < 9.
a(3) = 686 since S(14, 10) = 686 is such that A372720(686) <= 0, etc.
Table of first and some notable terms:
       n        q     i         a(n) a(n)/q  A372720(a(n))
  --------------------------------------------------------
       1        6    12         162   3^3         -2
       2       10     9         250   5^2         -1
       3       14    10         686   7^2         -2
       4       15    11        1875   5^3         -1
       5       21    13        7203   7^3         -3
       6       22    12        2662   11^2        -4
       7       26    13        4394   13^2        -5
       8       30    16         750   5^2          0
      82      210    51       26250   5^3        -11
    1061     2310    99      635250   5^2 * 11    -3
   15013    30030   222    25375350   5 * 13^2   -30
  268015   510510   338   679488810   11^3       -18
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000
• Michael De Vlieger, Log log scatterplot of a(n), n = 1..268015 in blue, and A120944(n)^3 in red.
• Michael De Vlieger, Diagram of S(6) = A033845 arranged such that k appears vertically in order of magnitude, with smallest at the bottom. Color function relates to A372720(k), with positive values from largest in light yellow grading to A372720(k) = 1 in orange, and negative values with smallest absolute value in dark blue to greatest in light blue. a(1) = 162 appears at right.
• Michael De Vlieger, Diagram of S(30) = A143207 arranged such that k appears vertically in order of magnitude, with smallest at the bottom. Color function is as above, but with A372720(k) = 0 in red. a(2) = 750 appears at left in red.

Crossrefs

Cf. A000005, A008479, A120944, A126706, A372720.

Programs

Mathematica

(* First, load function f from A162306 *)
Table[k = 1; s = f[n, n^3]; While[DivisorSigma[0, n*s[[k]]] - k > 0, k++]; s[[k]], {n, Select[Range[6, 120], And[SquareFreeQ[#], CompositeQ[#]] &]}]