## Pos+ integers (a,b); a + (a+1) + (a+2) + … + (b-1) + b = a*b

Pairs   $(a, \; b)$   of positive integers for which

$a \; + \; (a+1) \; + \; (a+2) \; + \; ... \; + \; (b-1) \; + \; b \; = \; a \, b$

3 + 4 + 5 + 6 = 18 =   $3\times 6$

15 + 16 + 17 + … + 34 + 35 = 525 =   $15\times 35$

85 + 86 + 87 + … + 203 + 204 = 17340 =   $85\times 204$

493 + 494 + 495 + … + 1189 = 586177 =   $493\times 1189$

2871 + 2872 + 2873 + … + 6930 = 19896030 =   $2871\times 6930$

16731 + 16732 + 16733 + … + 40391 = 675781821 =   $16731\times 40391$

97513 + 97514 + 97515 + … + 235416 = 22956120408 =   $97513\times 235416$

568345 + 568346 + 568347 + … + 1372105 = 779829016225 =   $568345\times 1372105$

3312555 + 3312556 + 3312557 + … + 7997214
= 26491211221770

=   $3312555\times 7997214$

19306983 + 19306984 + 19306985 + … + 46611179
= 899921240562957

=   $19306983\times 46611179$

112529341 + 112529342 + 112529343 + … + 271669860
= 30570830315362260

=   $112529341\times 271669860$

655869061 + 655869062 + 655869063 + … + 1583407981
= 1038508305678375841

=   $655869061\times 1583407981$

3822685023 + 3822685024 + 3822685025 + … + 9228778026
= 35278711540581704598

=   $3822685023\times 9228778026$

22280241075 + 22280241076 + 22280241077 + … + 53789260175
= 1198437683944896688125

=   $22280241075\times 53789260175$

Find a recursion relation.

math grad - Interest: Number theory
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### 2 Responses to Pos+ integers (a,b); a + (a+1) + (a+2) + … + (b-1) + b = a*b

1. paul says:

The first number’s in the sequences follow a recurrence of {7,-7,1}. The next number given the first 3 is as follows. given {3,15,85} we have (7 x 85) + (-7 x 15) + (1 x 3) = 493

The last number in the sequences follow a recrrence of {6,-1}, so given the first 2 numbers we get, (6 x 35) + (-1 x 6) = 204

There is also a recurrence for the resultant or (a x b) value, this is

{41,-246, 246, -41, 1}, so given {18,525,17340,586177, 19896030} we get

(1 x 18) + (-41 x 525) + (246 x 17340) + (-246 x 586177) + (41 x 19896030) = 255915594

Using those to find the next sequence gives

129858761425 + ….. + 313506783024 = 40711602541832856049200

here are the recurrence again

{7, -7, 1}..{6, -1}..{41,-246, 246, -41, 1}

Paul