cp's OEIS Frontend

Also numbers whose binary indices together with 0 define a Golomb ruler.

The k-th composition in standard order (row k of 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.

After a(1) = 0, first differs from A305611 at a(42) = 6, A305611(42) = 7.

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_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.

The standard order of compositions is given by A066099.

The k-th composition in standard order (row k of 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. - Gus Wiseman, Apr 03 2020

First differs from A181796 at a(180) = 9, A181796(180) = 10.

First differs from A335549 at a(90) = 7, A335549(90) = 8.

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

We define a (normal) 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 contiguously match a pattern P if there is a contiguous subsequence of S whose parts have the same relative order as P. For example, (3,1,1,3) contiguously matches (1,1,2) and (2,1,1) but not (2,1,2), (1,2,1), (1,2,2), or (2,2,1).

First differs from A353432 (the consecutive case) in having 0 and 53.

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 A353696 (the consecutive version) in having 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 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.

First differs from A353402 (the non-consecutive version) in lacking 53.

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.

Every sequence can be uniquely split into a sequence of non-overlapping runs. For example, the runs of (2,2,1,1,1,3,2,2) are ((2,2),(1,1,1),(3),(2,2)), with sums (4,3,3,4).

Two numbers n_1 and n_2 are called c-equivalent (n_1~n_2) if in the binary they have the same parts of the form 10...0 with k>=0 zeros up to a permutation of them. For example, 6~5, 14~13~11, 12~9.

Denote by (10...0)^k the concatenation k the same consecutive parts (10...0). By agreement, (10...0)^0 denotes the absence of the corresponding part in the binary of n. Let n contains k_i parts with i-1 zeros, i=1,2,... . Then n~concatenation((1)^k_1, (10)^k_2,(100)^k_3,...). The latter number is a(n). Thus a(n_1)=a(n_2) if and only if n_1~n_2. For example, since a(19)=28 which is in binary 11100, then the canonical representation of 19 is (1)^2[*](100), where [*] means concatenation. Analogously, since a(23)=30 which in binary 11110, then the canonical representation of 23 is (1)^3[*](10).

As a natural application, consider a notion of parts power divisor of canonical representation of n. We consider parts power divisors only of the form a(m).

If the canonical representation of n is a(n)=(1)^k_1[*](10)^k_2[*](100)^k_3[*]..., then number a(m) is a parts power divisor of a(n), iff a(m)=(1)^t_1[*](10)^t_2[*](100)^t_3[*]... with all t_i<=k_i. In particular, 0 (with all t_i=0) is parts power divisor of every a(n). From this it follows that the number of primes power divisors of a(n) is (k_1+1)*(k_2+1)*... This number is an upper estimate for A124771(n).