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panconsummate numbers
A number    is panconsummate if for every base  , there is a number    such that    divided by its sum of digits in base    gives  .

In other words, a number is panconsummate it is not inconsummate in any base.

Checking for this property is made easier by noting that a number    is always "consummate" in a base  .

For example, 5 is panconsummate because (a)    and  , (b)    and  , and (c)    and  . On the contrary, 62 is not panconsummate because in base 10 it does not exist a number    such that    = 62 (i.e., 62 is inconsummate).

The known panconsummate numbers are 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 18, 20, 21, 23, 24, 31, 34, 36, 37, 39, 40, 43, 45, 53, 54, 57, 59, 61, 69, 72, 73, 77, 78, 81, 85, 89, 91, 121, 127, 144, 166, 169, 211, 219, 231, 239, 257, 267, 271, 331, 337, 353, 361, 413, 481, 523, 571, 661, 721, 1093, 1291, 3097.

It there exist a further such number (which seems improbable), it must be greater than  .

Panconsummate numbers can also be... (you may click on names or numbers and on + to get more values)