cp's OEIS Frontend

A number k is called a c-perfect number if the sum of its proper c-divisors equals k.

For the definition of a c-divisor of an integer, see comment in A124771.

From Charlie Neder, Jan 17 2019: (Start)

Sequence in binary: 1011, 1000111, 11100010, 111001011100, 100001000001111, 100010111000111100011, 10010000000101000110011, 100001111110010100110011...

Next term > 10^7. (End)

If the canonical representation of n is A233569(n)=(1)^k_1[*](10)^k_2[*]...[*](10...0)^k_t, where [*] means concatenation, then we say that a number (1)^r_1[*](10)^r_2[*]...[*](10...0)^r_t is a parts power divisor of canonical representation of n, iff all r_i<=k_i.

Note that, by agreement, (10...0)^0 means the absence of the corresponding part.

The number of appearances of k is the number of compositions of k into numbers of the form 3^n and 2*3^n, A235684(k).

The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

First differs from the non-consecutive version A353431 in lacking 22318, corresponding to the binary word 101011100101110 and standard composition (2,2,1,1,3,2,1,1,2), whose run-lengths (2,2,1,1,2,1) are a subsequence but not a consecutive subsequence.

The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

If a collapse is an adding together of some partial run of an integer composition c, we say c is collapsible iff by some sequence of collapses it can be reduced to a single part. An example of such a sequence of collapses is (11132112) -> (332112) -> (33222) -> (6222) -> (66) -> (n), which shows that (11132112) is a collapsible composition of 12.

Every number in binary is a concatenation of parts of the form 10...0 with k>=0 zeros. For example, 5=(10)(1), 11=(10)(1)(1), 7=(1)(1)(1). We call d>0 is a c-divisor of m, if d consists of some consecutive parts of m which are following in natural order (from the left to the right) in m (cf. comment in A124771). Note that, to d=0 corresponds an empty set of parts. So it is natural to consider 0 as a c-divisor of every m. For example, 3=(1)(1) is a c-divisor of 23, since (1)(1) includes in 23=(10)(1)(1)(1) in a natural order. Analogously, 1,2,5,7,11,23 are c-divisors of 23. But 6=(1)(10) is not a c-divisor of 23.

c-GCD(k,m) is called maximal common c-divisor of k,m. For example, c-GCD(7,11)=3.

Two numbers k,m are called mutually c-prime one to another, if c-GCD(k,m)=0.

In particular, 0 is c-prime to 0 (cf. 1 is prime to 1). Therefore, a(0)=1. Besides, every positive integer is c-prime to 0.

We define a pattern to be a finite sequence covering an initial interval of positive integers. Patterns are counted by A000670 and ranked by A333217. A sequence S is said to match a pattern P if there is a not necessarily contiguous subsequence of S whose parts have the same relative order as P. For example, (3,1,1,3) matches (1,1,2), (2,1,1), and (2,1,2), but avoids (1,2,1), (1,2,2), and (2,2,1).