A130733 Numbers whose square can be expressed as a+b*c, with a,b,c in geometric sequence.

3, 102, 130, 312, 759, 2496, 2706, 3465, 6072, 6111, 8424, 14004, 16005, 36897, 37156, 92385, 98640, 112032, 117708, 128040, 351260, 378108, 740050, 1346400, 1371900, 1898130, 3998607, 5986575, 6082065, 6631596, 6741214, 7692804

Offset: 1

Author

Graeme McRae, Jul 05 2007

Keywords

• nonn

Comments

This sequence was inspired by a puzzle question that asked for all squares under a trillion that can be expressed as either a+bc or b+ac where a,b,c are in increasing geometric progression.

Examples

			a(1)=3 because 3^2=1+2*4 and 1,2,4 are in geometric sequence
a(2)=102 because 102^2=36+72*144 and 36,72,144 are in geometric sequence
a(3)=130 because 130^2=25+75*225 and 25,75,225 are in geometric sequence
a(4)=312 because 312^2=8+92*1058 and ...
a(5)=759 because 759^2=81+360*1600
a(6)=2496 because 2496^2=512+1472*4232
a(7)=2706 because 2706^2=1936+2420*3025
a(8)=3465 because 3465^2=1225+2450*4900
a(9)=6072 because 6072^2=5184+5760*6400
a(10)=6111 because 6111^2=3969+5292*7056
a(11)=8424 because 8424^2=5832+7452*9522
a(12)=14004 because 14004^2=432+4392*44652
a(13)=16005 because 16005^2=1089+6534*39204
a(14)=36897 because 36897^2=21609+30870*44100
a(15)=37156 because 37156^2=12544+25872*53361
a(16)=92385 because 92385^2=50625+75600*112896
a(17)=98640 because 98640^2=50625+78975*123201
a(18)=112032 because 112032^2=27648+70272*178608
a(19)=117708 because 117708^2=41616+83232*166464
a(20)=128040 because 128040^2=69696+104544*156816
a(21)=351260 because 351260^2=67600+202800*608400
a(22)=378108 because 378108^2=314928+355752*401868
a(23)=740050 because 740050^2=521284+658464*831744
No other numbers smaller than a million have squares that can be expressed this way.
Contribution from _Donovan Johnson_, Jul 30 2010: (Start)
a(24)=1346400 because 1346400^2=135000+625500*2898150
a(25)=1371900 because 1371900^2=10000+266000*7075600
a(26)=1898130 because 1898130^2=6084+279864*12873744
a(27)=3998607 because 3998607^2=1413721+2827442*5654884
a(28)=5986575 because 5986575^2=1157625+3461850*10352580
a(29)=6082065 because 6082065^2=4348377+5438466*6801828
a(30)=6631596 because 6631596^2=1944+440532*99829446
a(31)=6741214 because 6741214^2=334084+2476152*18352656
a(32)=7692804 because 7692804^2=444528+2974104*19898172
(End)
		

Extensions

Added word: 'increasing'. The original puzzle was expressed as a modulo operation, the expression was 'remainder + quotient * divisor', where the remainder is necessarily smaller than the divisor, implying an increasing sequence. Counterexample if 'increasing' is not specified: a=8, b=4, c=2. a+b*c = 16 = 4^2; 4 is not in sequence A130733 - James Cunnane (james.cunnane(AT)gmail.com), Jun 29 2010

a(24)-a(32) from Donovan Johnson, Jul 30 2010