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A289385
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Nonprime numbers k such that the sum of the divisors of k^2 is of the form m^2 + m + 1.
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1
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1, 4, 6, 22, 48, 64, 390, 429, 638, 682, 880, 2013, 2310, 2320, 2968, 2970, 3625, 4692, 5696, 5865, 7772, 9213, 9715, 9944, 10778, 11254, 12963, 13039, 13704, 15180, 23370, 27236, 27556, 34045, 34445, 36141, 44336, 53136, 54376, 58548, 73185, 78952, 105464
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OFFSET
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1,2
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COMMENTS
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The corresponding values of m are given by the sequence {0, 5, 9, 30, 81, 90, 718, 562, 900, 961, 1451, ...} containing the subsequence of perfect squares {0, 9, 81, 900, 961, 4624, 8100, ...}.
If k is prime, the sum of the divisors of k^2 is the trivial value k^2 + k + 1.
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LINKS
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EXAMPLE
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22 is in the sequence because sigma(22^2) = 931 = 30^2 + 30 + 1.
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MAPLE
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with(numtheory):nn:=10^5:
for n from 1 to nn do:
if type(n, prime)=false
then
y:=sqrt(4*sigma(n^2)-3):
z :=(-1+y)/2:
if y=floor(y) and z=floor(z) then printf(`%d, `, n):
else
fi:fi:
od:
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MATHEMATICA
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fQ[n_] := ! PrimeQ@ n && Block[{sd = DivisorSigma[1, n^2]}, y = Sqrt[4sd -3]; z = (y -1)/2; y == Floor@ y && z == Floor@ z]; Select[Range@ 125491, fQ] (* Robert G. Wilson v, Jul 05 2017 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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