A117949 Index of pentagonal numbers whose sum of divisors is square.
1, 4, 7, 12, 21, 23, 27, 31, 71, 79, 89, 151, 168, 199, 223, 232, 239, 263, 311, 324, 336, 345, 359, 390, 463, 479, 497, 540, 599, 743, 751, 823, 858, 863, 911, 991, 1031, 1063, 1103, 1151, 1302, 1303, 1343, 1399, 1471, 1540, 1583, 1687, 1759, 1802, 1823
Offset: 1
Examples
a(1) = 1 because sigma(1*(3*1-1)/2) = 1 = 1^2. a(2) = 4 because sigma(4*(3*4-1)/2) = 36 = 6^2. a(3) = 7 because sigma(7*(3*7-1)/2) = 144 = 12^2. a(4) = 12 because sigma(12*(3*12-1)/2) = 576 = 24^2. a(5) = 21 because sigma(21*(3*21-1)/2) = 1024 = 32^2. a(6) = 23 because sigma(23*(3*23-1)/2) = 1296 = 36^2. a(7) = 27 because sigma(27*(3*27-1)/2) = 3600 = 60^2. a(8) = 31 because sigma(31*(3*31-1)/2) = 2304 = 48^2. a(9) = 71 because sigma(71*(3*71-1)/2) = 11664 = 108^2.
Links
- Amiram Eldar, Table of n, a(n) for n = 1..10000
Programs
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Maple
with(numtheory): select(n-> sqrt(sigma(n*(3*n-1)/2))::integer, [$1..2200])[]; # Emeric Deutsch, Apr 06 2006
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Mathematica
s = {}; Do[If[IntegerQ @ Sqrt @ DivisorSigma[1, (3 n - 1)*n/2], AppendTo[s, n]], {n, 1, 2000}]; s (* Amiram Eldar, Aug 17 2019 *) Position[DivisorSigma[1,PolygonalNumber[5,Range[2000]]],?(IntegerQ[ Sqrt[ #]]&)]//Flatten (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale, Oct 23 2020 *) -
PARI
isok(n) = issquare(sigma(n*(3*n-1)/2)); \\ Michel Marcus, Aug 17 2019
Extensions
More terms from Emeric Deutsch, Apr 06 2006
a(0) removed by Amiram Eldar, Aug 17 2019
Comments