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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.

%I A373737 #7 Jul 07 2024 20:55:45
%S A373737 162,250,686,1875,7203,2662,4394,750,3993,578,12005,722,6591,2058,
%T A373737 1058,14739,73205,20577,1682,1922,142805,5346,36501,3430,2738,102487,
%U A373737 6318,3362,417605,3698,73167,199927,89373,4418,651605,5202,25725,5618,13310,151959,6498
%N 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.
%C A373737 Numbers k whose position i in S(n) is such that tau(k) <= i, i.e., that A372720(k) is not positive.
%C A373737 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.
%C A373737 a(n) is in A126706.
%C A373737 Conjecture: a(n) <= s*gpf(s)^floor(log_gpf(s) s^2), where gpf = A006530.
%H A373737 Michael De Vlieger, <a href="/A373737/b373737.txt">Table of n, a(n) for n = 1..10000</a>
%H A373737 Michael De Vlieger, <a href="/A373737/a373737.png">Log log scatterplot of a(n)</a>, n = 1..268015 in blue, and A120944(n)^3 in red.
%H A373737 Michael De Vlieger, <a href="/A373737/a373737_1.png">Diagram of S(6) = A033845</a> 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.
%H A373737 Michael De Vlieger, <a href="/A373737/a373737_2.png">Diagram of S(30) = A143207</a> 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.
%e A373737 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.
%e A373737 a(2) = 250 since S(10, 9) = 250 gives tau(250) = 8, and 8 < 9.
%e A373737 a(3) = 686 since S(14, 10) = 686 is such that A372720(686) <= 0, etc.
%e A373737 Table of first and some notable terms:
%e A373737        n        q     i         a(n) a(n)/q  A372720(a(n))
%e A373737   --------------------------------------------------------
%e A373737        1        6    12         162   3^3         -2
%e A373737        2       10     9         250   5^2         -1
%e A373737        3       14    10         686   7^2         -2
%e A373737        4       15    11        1875   5^3         -1
%e A373737        5       21    13        7203   7^3         -3
%e A373737        6       22    12        2662   11^2        -4
%e A373737        7       26    13        4394   13^2        -5
%e A373737        8       30    16         750   5^2          0
%e A373737       82      210    51       26250   5^3        -11
%e A373737     1061     2310    99      635250   5^2 * 11    -3
%e A373737    15013    30030   222    25375350   5 * 13^2   -30
%e A373737   268015   510510   338   679488810   11^3       -18
%t A373737 (* First, load function f from A162306 *)
%t A373737 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[#]] &]}]
%Y A373737 Cf. A000005, A008479, A120944, A126706, A372720.
%K A373737 nonn,easy
%O A373737 1,1
%A A373737 _Michael De Vlieger_, Jun 24 2024