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A272442
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Numbers n such that the average of the positive divisors of n^2 is a square.
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0
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1, 313, 4077, 42217, 75848, 1276101, 2468936, 8105257, 13213921, 23740424, 76136648, 172118709, 249948601, 296474857, 309232296, 772776968, 1236329761, 1254551521, 2288805793, 2372141576, 2410951561, 2536945441, 3202075016, 10065852072
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OFFSET
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1,2
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COMMENTS
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The values of n are 1, 313, 3^3*151, 7*37*163, 2^3*19*499, 3^3*151*313, 2^3*19*37*439, 37*439*499, ...
Corresponding n^2 values are 1, 97969, 16621929, 1782275089, ...
The next two primes in the sequence (after 313 and 2288805793) are 86136917171761 and 23666840016182721134955337. - Giovanni Resta, Apr 26 2017
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LINKS
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EXAMPLE
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313 is a term because the divisors of 313^2 are 1, 313 and 313^2. Average of 1, 313 and 313^2 is 32761 that is 181^2.
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MATHEMATICA
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Select[Range[10^5], IntegerQ@ Sqrt@ Mean@ Divisors[#^2] &] (* Michael De Vlieger, Apr 29 2016 *)
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PROG
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(PARI) lista(nn) = for(n=1, nn, if(sigma(n^2) % numdiv(n^2) == 0 && issquare(sigma(n^2)/numdiv(n^2)), print1(n, ", ")));
(PARI) is(n)=my(f=factor(n), s, d); f[, 2]*=2; s=sigma(f); d=numdiv(f); s%d==0 && issquare(s/d) \\ Charles R Greathouse IV, Apr 29 2016
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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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