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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.
1

%I #7 Jul 07 2024 20:55:45

%S 162,250,686,1875,7203,2662,4394,750,3993,578,12005,722,6591,2058,

%T 1058,14739,73205,20577,1682,1922,142805,5346,36501,3430,2738,102487,

%U 6318,3362,417605,3698,73167,199927,89373,4418,651605,5202,25725,5618,13310,151959,6498

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

%C 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 a(n) is in A126706.

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

%H Michael De Vlieger, <a href="/A373737/b373737.txt">Table of n, a(n) for n = 1..10000</a>

%H 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 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 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 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 a(2) = 250 since S(10, 9) = 250 gives tau(250) = 8, and 8 < 9.

%e a(3) = 686 since S(14, 10) = 686 is such that A372720(686) <= 0, etc.

%e Table of first and some notable terms:

%e n q i a(n) a(n)/q A372720(a(n))

%e --------------------------------------------------------

%e 1 6 12 162 3^3 -2

%e 2 10 9 250 5^2 -1

%e 3 14 10 686 7^2 -2

%e 4 15 11 1875 5^3 -1

%e 5 21 13 7203 7^3 -3

%e 6 22 12 2662 11^2 -4

%e 7 26 13 4394 13^2 -5

%e 8 30 16 750 5^2 0

%e 82 210 51 26250 5^3 -11

%e 1061 2310 99 635250 5^2 * 11 -3

%e 15013 30030 222 25375350 5 * 13^2 -30

%e 268015 510510 338 679488810 11^3 -18

%t (* First, load function f from A162306 *)

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

%K nonn,easy

%O 1,1

%A _Michael De Vlieger_, Jun 24 2024