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A136404
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Square numbers with more divisors than any smaller square number.
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6
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1, 4, 16, 36, 144, 576, 900, 3600, 14400, 32400, 44100, 129600, 176400, 705600, 1587600, 2822400, 6350400, 21344400, 57153600, 85377600, 192099600, 341510400, 768398400, 3073593600, 6915585600, 12294374400, 14428814400, 32464832400, 57715257600, 129859329600
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OFFSET
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1,2
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COMMENTS
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Being the square of a number in A002182 is neither necessary nor sufficient.
Conjecture: square roots of the terms of this sequence are the same terms as A126098.
Records for largest exponents occur at: 1, 2, 5, 15, 25, 35, 200, 203
Least k such that a(k) divides prime(i)^4: 5, 10, 34, 104, 302
Based on these exponents I made the following dataset:
primorials <= 10^200 (92 such numbers).
Then made products of primorials <= 10^200 where the exponent of prime(11) is at most 2. Then searched records here. The b-file is checking A025487 squared checked. (End)
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LINKS
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EXAMPLE
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900 qualifies because 576 has only 21 divisors and 900 has 27. 1296 does not because 1296 has only 25 divisors as opposed to the 27 of the smaller 900.
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MAPLE
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a := 0 : for n from 1 to 1000000 do ndvs := numtheory[tau](n^2) ; if ndvs > a then printf("%d, ", n^2) ; a := ndvs ; fi ; od: # R. J. Mathar, Apr 04 2008
with(numtheory): a:=proc(n) if max(seq(tau(j^2), j=1..n-1))<tau(n^2) then n^2 else end if end proc: seq(a(n), n=1..10000); # Emeric Deutsch, Apr 04 2008
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MATHEMATICA
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With[{s = Array[DivisorSigma[0, #^2] &, 10^6]}, Map[FirstPosition[s, #][[1]]^2 &, Union@ FoldList[Max, s]]] (* Michael De Vlieger, Oct 15 2018 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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