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A number that has the same number of digits as the number of digits in its prime factorization, including exponents but excluding exponents equal to 1. A046758 Extravagant numbers
For example the five compositions of 5 into distinct terms are: 5; 4 + 1; 3 + 2; 2 + 3; 1 + 4. Compare this with the three partitions of 5 into distinct terms: 5; 4 + 1; 3 + 2. Note that the ancient Sanskrit sages discovered many years before Fibonacci that the number of compositions of any natural number n as the sum of 1's and 2's is the nth ...
Suppose one wants to determine the 5-combination at position 72. The successive values of () for n = 4, 5, 6, ... are 0, 1, 6, 21, 56, 126, 252, ..., of which the largest one not exceeding 72 is 56, for n = 8. Therefore c 5 = 8, and the remaining elements form the 4-combination at position 72 − 56 = 16. The successive values of () for n = 3 ...
This is the minimum number of characters needed to encode a 32 bit number into 5 printable characters in a process similar to MIME-64 encoding, since 85 5 is only slightly bigger than 2 32. Such method is 6.7% more efficient than MIME-64 which encodes a 24 bit number into 4 printable characters. 89
A list of articles about numbers (not about numerals). Topics include powers of ten, notable integers, prime and cardinal numbers, and the myriad system.
The number of such strings is the number of ways to place 10 stars in 13 positions, () = =, which is the number of 10-multisubsets of a set with 4 elements. Bijection between 3-subsets of a 7-set (left) and 3-multisets with elements from a 5-set (right).
Numerical 3-dimensional matching is an NP-complete decision problem. It is given by three multisets of integers, and , each containing elements, and a bound .The goal is to select a subset of such that every integer in , and occurs exactly once and that for every triple (,,) in the subset + + = holds.
In plain words, e.g., the first congruence means that If a number is 4 more than a multiple of 5, i.e. it is in the sequence 4, 9, 14, 19, 24, 29, . . . then the number of its partitions is a multiple of 5. Later other congruences of this type were discovered, for numbers and for Tau-functions.