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A373737
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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.
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1
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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
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OFFSET
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1,1
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COMMENTS
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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.
Conjecture: a(n) <= s*gpf(s)^floor(log_gpf(s) s^2), where gpf = A006530.
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LINKS
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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.
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EXAMPLE
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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:
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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
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MATHEMATICA
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(* 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[#]] &]}]
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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