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A117949 Index of pentagonal numbers whose sum of divisors is square.

%I A117949 #25 Sep 13 2024 00:39:44
%S A117949 1,4,7,12,21,23,27,31,71,79,89,151,168,199,223,232,239,263,311,324,
%T A117949 336,345,359,390,463,479,497,540,599,743,751,823,858,863,911,991,1031,
%U A117949 1063,1103,1151,1302,1303,1343,1399,1471,1540,1583,1687,1759,1802,1823
%N A117949 Index of pentagonal numbers whose sum of divisors is square.
%C A117949 n such that A117948(n) is in A000290.
%H A117949 Amiram Eldar, <a href="/A117949/b117949.txt">Table of n, a(n) for n = 1..10000</a>
%e A117949 a(1) = 1 because sigma(1*(3*1-1)/2) = 1 = 1^2.
%e A117949 a(2) = 4 because sigma(4*(3*4-1)/2) = 36 = 6^2.
%e A117949 a(3) = 7 because sigma(7*(3*7-1)/2) = 144 = 12^2.
%e A117949 a(4) = 12 because sigma(12*(3*12-1)/2) = 576 = 24^2.
%e A117949 a(5) = 21 because sigma(21*(3*21-1)/2) = 1024 = 32^2.
%e A117949 a(6) = 23 because sigma(23*(3*23-1)/2) = 1296 = 36^2.
%e A117949 a(7) = 27 because sigma(27*(3*27-1)/2) = 3600 = 60^2.
%e A117949 a(8) = 31 because sigma(31*(3*31-1)/2) = 2304 = 48^2.
%e A117949 a(9) = 71 because sigma(71*(3*71-1)/2) = 11664 = 108^2.
%p A117949 with(numtheory): select(n-> sqrt(sigma(n*(3*n-1)/2))::integer, [$1..2200])[]; # _Emeric Deutsch_, Apr 06 2006
%t A117949 s = {}; Do[If[IntegerQ @ Sqrt @ DivisorSigma[1, (3 n - 1)*n/2], AppendTo[s, n]], {n, 1, 2000}]; s (* _Amiram Eldar_, Aug 17 2019 *)
%t A117949 Position[DivisorSigma[1,PolygonalNumber[5,Range[2000]]],_?(IntegerQ[ Sqrt[ #]]&)]//Flatten (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale_, Oct 23 2020 *)
%o A117949 (PARI) isok(n) = issquare(sigma(n*(3*n-1)/2)); \\ _Michel Marcus_, Aug 17 2019
%Y A117949 Cf. A000203, A000217, A000290, A000326, A074285, A083675, A117948.
%K A117949 easy,nonn
%O A117949 1,2
%A A117949 _Jonathan Vos Post_, Apr 04 2006
%E A117949 More terms from _Emeric Deutsch_, Apr 06 2006
%E A117949 a(0) removed by _Amiram Eldar_, Aug 17 2019