Biquanimous Numbers

The number associated with my diurnal age today, 27190, has the interesting property that it counts the number of five digit biquanimous numbers. This covers the range from 10000 to 99999 which means that about 27% of five digit numbers are biquanimous, a term used to describe numbers whose digits can be split into two groups with equal sums. By an odd coincidence, the number following 27190 is an example of such a number because:$$27191 \rightarrow \underbrace{2+7+1}_{\text{ sums to } 10} \text{ and } \underbrace{9+1}_{\text{ sums to } 10}$$Now just to repeat: 27190 is not biquanimous but it does count the number of five digit biquanimous numbers. These totals of \(n\)-digit biquanimous numbers constitute OEIS A065086:

A065086

Number of \(n\)-digit biquanimous numbers in base 10 not allowing leading zeros.

The initial members of the sequence are:

1, 9, 126, 1920, 27190, 347168, 3990467, 42744527, 440764556, 4464045276, 44863859589, 449488519847, 4498059105204, 44992451829730, 449969622539954, 4499873022468708, 44999449703306768, 449997540235466340, 4499988731927483569, 44999947410278947807

It can be seen that the limit as \(n \rightarrow \infty \) seems to be 45% but in fact, if leading zeros are allowed, the limit is 0.5 according to this source. The biquanimous numbers, also known as biquams, are listed in OEIS A064544:

A064544

Biquanimous numbers (or biquams): group the digits into two pieces (not necessarily equal or in order) with the same sum.

The initial members are:

0, 11, 22, 33, 44, 55, 66, 77, 88, 99, 101, 110, 112, 121, 123, 132, 134, 143, 145, 154, 156, 165, 167, 176, 178, 187, 189, 198, 202, 211, 213, 220, 224, 231, 235, 242, 246, 253, 257, 264, 268, 275, 279, 286, 297, 303, 312, 314, 321, 325, 330, 336, 341, 347, 352, 358

This approach is similar to totaling the odd and even numbers and seeing if they balance. Numbers that do constitute OEIS  A036301:

A036301

Numbers whose sum of even digits and sum of odd digits are equal.

The initial members are:

0, 112, 121, 134, 143, 156, 165, 178, 187, 211, 314, 336, 341, 358, 363, 385, 413, 431, 516, 538, 561, 583, 615, 633, 651, 718, 781, 817, 835, 853, 871, 1012, 1021, 1034, 1043, 1056, 1065, 1078, 1087, 1102, 1120, 1201, 1210, 1223, 1232, 1245, 1254, 1267, 1276, 1289, 1298 

It can be noted that all such numbers are by definition biquanimous because their digits can be split into two groups with equal sums. For example:$$1298 \rightarrow \underbrace{1+9}_{\text{sums to }10 } \text{ and } \underbrace{2+8}_{ \text{sums to } 10}$$