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A318807
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Numbers whose sum of squarefree divisors and sum of nonsquarefree divisors are both a perfect square.
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1
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1, 3, 9, 22, 27, 66, 70, 88, 94, 115, 119, 170, 198, 210, 214, 217, 264, 265, 280, 282, 310, 322, 345, 357, 376, 382, 385, 497, 510, 517, 527, 594, 630, 642, 651, 679, 680, 710, 729, 742, 745, 782, 795, 840, 846, 856, 862, 889, 930, 935, 966, 970, 1035, 1066
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OFFSET
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1,2
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COMMENTS
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Let s be the sum of the squarefree divisors of a number m. The sequence lists the numbers m such that s and sigma(m) - s are both a perfect square.
The corresponding pairs of squares (s, sigma(m) - s) are (1, 0), (4, 0), (4, 9), (36, 0), (4, 36), (144, 0), (144, 0), (36, 144), (144, 0), (144, 0), (144, 0), (324, 0), (144, 324), ...
The subsequence b(n) where s and sigma(m) - s are strictly positive begins with 9, 27, 88, 198, 264, 280, 376, 594, 630, ... b(n) is not squarefree (subsequence of A013929).
The subsequence c(n) where the ratio r = (sigma(a(n)) - s)/s is integer begins with 27, 88, 264, 280, 376, 594, 680, 840, 856, 1128, 1240, ... and the corresponding r are 3^2, 2^2, 2^2, 2^2, 2^2, 3^2, 2^2, 2^2, 2^2, 2^2, 2^2, 2^2, 2^2, 5^2, 3^2, 2^2, 7^2, 3^2, 2^2, 11^2, ... It is conjectured that r belongs to A001248.
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LINKS
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EXAMPLE
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27 is in the sequence because A048250(27) = 4 and A162296(27) = 36 are both a perfect square.
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MAPLE
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filter:= proc(n) local F, SF, NSF, t;
F:= ifactors(n)[2];
SF:= mul(1+t[1], t=F);
if not issqr(SF) then return false fi;
NSF:= mul((1-t[1]^(1+t[2]))/(1-t[1]), t=F) - SF;
issqr(NSF);
end proc:
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MATHEMATICA
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lst={}; Do[If[IntegerQ[Sqrt[Total[Select[Divisors[n], SquareFreeQ]]]]&&IntegerQ[Sqrt[DivisorSigma[1, n]-Total[Select[Divisors[n], SquareFreeQ]]]], AppendTo[lst, n]], {n, 1100}]; lst
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PROG
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(PARI) isok(n) = {my(sd=sumdiv(n, d, issquarefree(d)*d)); issquare(sd) && issquare(sigma(n) - sd); } \\ Michel Marcus, Sep 04 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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STATUS
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approved
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