I came across a mathematical term today that I hadn't heard of before. The term is "aliquant" defined as follows by contrasting it to similar sounding "aliquot" (source):
Webster defines 'aliquot' as something that contained an exact number of times in something else or to divide into equal parts.
Notice the word "equal". An example being 5 is an aliquot part of 15.
The term 'aliquant', however, is slightly different. Defined as being a part of a number or quantity, but not dividing it without leaving a remainder. An example being 5 is an aliquant part of 16.
A098743: number of partitions of \(n\) into aliquant parts (i.e., parts that do not divide \(n\)).
The initial members of the sequence are:
1, 0, 0, 0, 0, 1, 0, 3, 1, 3, 3, 13, 1, 23, 10, 11, 9, 65, 8, 104, 14, 56, 66, 252, 10, 245, 147, 206, 77, 846, 35, 1237, 166, 649, 634, 1078, 60, 3659, 1244, 1850, 236, 7244, 299, 10086, 1228, 1858, 4421, 19195, 243, 17660, 3244, 12268, 4039, 48341, 1819, 27675
The last member shown above, 27675, is my diurnal age today and corresponds to \(n=55\). To find the aliquant parts, simply remove the divisors of the number. For example, 55 has divisors of 1, 5, 11 and 55 and so the number of aliquant parts is 51. The list is as follows:
2, 3, 4, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54
Note that these are different to the numbers that contribute to the total of the totient of a number, where the numbers are coprime. The totient for 55 is 40, made up of the following numbers.
1, 2, 3, 4, 6, 7, 8, 9, 12, 13, 14, 16, 17, 18, 19, 21, 23, 24, 26, 27, 28, 29, 31, 32, 34, 36, 37, 38, 39, 41, 42, 43, 46, 47, 48, 49, 51, 52, 53, 54
