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A130733 Numbers whose square can be expressed as a+b*c, with a,b,c in geometric sequence. 1
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

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.

LINKS

Table of n, a(n) for n=1..32.

EXAMPLE

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)

CROSSREFS

Sequence in context: A142416 A110625 A108220 * A037062 A290027 A224817

Adjacent sequences:  A130730 A130731 A130732 * A130734 A130735 A130736

KEYWORD

nonn

AUTHOR

Graeme McRae, Jul 05 2007

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

STATUS

approved

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Last modified January 26 13:09 EST 2022. Contains 350598 sequences. (Running on oeis4.)