## (a,b) such that d(a)=8, d(b)=18, |a-b|=28

$d(n)$   is the number of positive divisors of   $n$,   including 1 and   $n$   itself.
$\sigma(n)$   is the sum of the positive divisors of   $n$,   including 1 and   $n$   itself
$s(n)$   is the sum of the proper divisors of   $n$,   which does not include   $n$   itself;
that is,   $s(n) = \sigma(n) - n$

To find   (a, b)   such that
a   has   8 divisors – including the divisor   8
b   has   18   divisors – including the divisor   18
|a – b| = 28

here are the first few examples,

152    |   1,2,4,8,19,38,76,152
180    |   1,2,3,4,5,6,9,10,12,15,18,20,30,36,45,60,90,180

180 – 152 = 28

424    |   1,2,4,8,53,106,212,424
396    |   1,2,3,4,6,9,11,12,18,22,33,36,44,66,99,132,198,396

424 – 396 = 28

584    |   1,2,4,8,73,146,292,584
612    |   1,2,3,4,6,9,12,17,18,34,36,51,68,102,153,204,306,612

612 – 584 = 28

712    |   1,2,4,8,89,178,356,712
684    |   1,2,3,4,6,9,12,18,19,36,38,57,76,114,171,228,342,684

712 – 684 = 28

856    |   1,2,4,8,107,214,428,856
828    |   1,2,3,4,6,9,12,18,23,36,46,69,92,138,207,276,414,828

856 – 828 = 28

Find other pairs   (a, b)

math grad - Interest: Number theory
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### One Response to (a,b) such that d(a)=8, d(b)=18, |a-b|=28

1. paul says:

Here are those others up to 5000

1016 , {1,2,4,8,127,254,508,1016}
1044 , {1,2,3,4,6,9,12,18,29,36,58,87,116,174,261,348,522,1044}
1044 – 1016 = 28

1304 , {1,2,4,8,163,326,652,1304}
1332 , {1,2,3,4,6,9,12,18,36,37,74,111,148,222,333,444,666,1332}
1332 – 1304 = 28

1448 , {1,2,4,8,181,362,724,1448}
1476 , {1,2,3,4,6,9,12,18,36,41,82,123,164,246,369,492,738,1476}
1476 – 1448 = 28

1548 , {1,2,3,4,6,9,12,18,36,43,86,129,172,258,387,516,774,1548}
1576 {1,2,4,8,197,394,788,1576}
1576 – 1548 = 28

2124 , {1,2,3,4,6,9,12,18,36,59,118,177,236,354,531,708,1062,2124}
2152 {1,2,4,8,269,538,1076,2152}
2152 – 2124 = 28

2168 , {1,2,4,8,271,542,1084,2168}
2196 , {1,2,3,4,6,9,12,18,36,61,122,183,244,366,549,732,1098,2196}
2196 – 2168 = 28

2844 , {1,2,3,4,6,9,12,18,36,79,158,237,316,474,711,948,1422,2844}
2872 {1,2,4,8,359,718,1436,2872}
2872 – 2844 = 28

3176 , {1,2,4,8,397,794,1588,3176}
3204 , {1,2,3,4,6,9,12,18,36,89,178,267,356,534,801,1068,1602,3204}
3204 – 3176 = 28

3464 , {1,2,4,8,433,866,1732,3464}
3492 , {1,2,3,4,6,9,12,18,36,97,194,291,388,582,873,1164,1746,3492}
3492 – 3464 = 28

3708 , {1,2,3,4,6,9,12,18,36,103,206,309,412,618,927,1236,1854,3708}
3736 {1,2,4,8,467,934,1868,3736}
3736 – 3708 = 28

3896 , {1,2,4,8,487,974,1948,3896}
3924 , {1,2,3,4,6,9,12,18,36,109,218,327,436,654,981,1308,1962,3924}
3924 – 3896 = 28

4716 , {1,2,3,4,6,9,12,18,36,131,262,393,524,786,1179,1572,2358,4716}
4744 {1,2,4,8,593,1186,2372,4744}
4744 – 4716 = 28

4904 , {1,2,4,8,613,1226,2452,4904}
4932 , {1,2,3,4,6,9,12,18,36,137,274,411,548,822,1233,1644,2466,4932}
4932 – 4904 = 28

Paul.