the sum of their proper divisors minus
the product of their digits equals the number
The numbers are 198, 4172, 7144 and 27824 and they are the initial members of OEIS A331098. Figure 1 shows the details:
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Figure 1: permalink |
Here is a longer list of the sequence's members:
198, 4172, 7144, 27824, 72212, 111126, 1111134, 1113114, 1131114, 7121212, 11131122, 13111122, 33550336, 111711124, 1111113114, 1111212172, 1113111114, 2111211172, 7111211212, 8589869056, 11112117212, 11113111122, 11121121172, 11711121212, 13111111122, 17112111212
The perfect numbers containing a 0 digit are all in this sequence (33550336 and 8589869056 are shown above).
What if the change the property so that we have:
the sum of their proper divisors minus
the sum of their digits equals the number
In this case, the numbers are more numerous. In the range up to 40000 we have (permalink):
20, 66, 138, 174, 246, 282, 318, 354, 426, 534, 606, 642, 822, 1038, 1074, 1146, 1182, 1362, 1434, 1506, 1542, 1614, 1902, 2082, 2118, 2154, 2334, 2406, 2514, 2802, 3018, 3054, 3126, 3342, 3414, 3522, 3702, 4062, 4206, 4314, 5034, 5142, 5322, 6114, 7122, 7232, 7302, 8202, 8415, 10182, 10254, 10326, 10722, 11082, 11226, 11262, 11334, 11406, 11442, 12018, 12162, 12234, 12414, 12522, 13242, 13422, 13602, 14034, 14106, 14142, 14502, 16122, 17022, 19002, 20082, 20154, 20226, 20334, 20442, 20802, 21162, 21234, 21342, 21702, 22026, 22062, 22206, 22314, 22602, 23106, 23502, 24006, 24042, 24114, 25302, 26022, 27102, 30018, 30054, 30126, 30234, 30306, 30522, 31026, 31134, 31422, 31815, 32106, 32322, 32502, 33006, 33042, 33114, 34014, 34302, 35202
For example, 20 has a sum of proper divisors equal to 22 and its sum of digits is 2, thus 22 - 2 = 20. These numbers form OEIS A331096. Most of the members are even numbers but 8415 and 31815 are odd.
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