Primorial Number Base Revisited

On the 14th February 2021, now over four years ago, I created a post on this blog about the Primorial Number System. Since then, I've thought very little about it but today's number (associated with my diurnal age) reminded me once again of this number system. The number is \( \textbf{27718} \) and it is a member of OEIS A333703:

A333703   Numbers \(k \)such that \(k\) divides the sum of digits in primorial base of all numbers from \(1\) to \(k\).

The numbers that satisfy up to 40000 are:

1, 2, 10, 22, 58, 62, 63, 64, 66, 67, 68, 118, 178, 418, 838, 1258, 1264, 1265, 1277, 1278, 1678, 2098, 4618, 9238, 10508, 10509, 10510, 10512, 10513, 10514, 13858, 14704, 14754, 18478, 23098, 23102, 23276, 27718

Table 1 shows the numbers from OEIS A333703 together with their primorial base equivalents and the progressive totals of the digits of the all the primorial numbers up and including each number. The primorial base representation I've employed here uses the base 10 digits (0 to 9) together with a space as a separator (although colons are more commonly used). However, for numbers in the range up to 40000 that I use the base 12 system using the additional digits A for 10 and B for 11 are sufficient so that concatenation of the "placeholders" does not produce any ambiguity. The primorial number then looks like a normal base 12 number which produces an ambiguity in itself.

Table 1: permalink

Table 2 shows the numbers together with their corresponding progressive totals and the results when these totals are divided by the corresponing number.

Table 2: permalink

The next number after 22718 is 60058 so I won't be around to see that. For more information see this source. I started this blog by referring to my diurnal age on the 21st February 2025 (27718) but my diurnal age on the very next day (\( \textbf{27719} \)) also has a property that connects it to the primorial number base.

A343048   a(\(n\)) is the least number whose sum of digits in primorial base equals \(n\).

The members of this sequence up to 40000 are (permalink):

0, 1, 3, 5, 11, 17, 23, 29, 59, 89, 119, 149, 179, 209, 419, 629, 839, 1049, 1259, 1469, 1679, 1889, 2099, 2309, 4619, 6929, 9239, 11549, 13859, 16169, 18479, 20789, 23099, 25409, 27719, 30029

Table 3 shows the increasing values of \(n\):

Table 3: permalink

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:$$1298 \rightarrow \underbrace{1+9}_{\text{sums to }10 } \text{ and } \underbrace{2+8}_{ \text{sums to } 10}$$