cp's OEIS Frontend

From Vladimir Shevelev, Dec 11 2014: (Start)

Or, sum of parts of the form 10...0 with nonnegative number of zeros in binary representation of n as the corresponding powers of 2. For example, n=50 in binary is a concatenation of parts (1)(100)(10). Then a(50)=1+4+2=7.

Every positive number k occurs a finite number of times, such that the position of the last appearance of k is 2^k-1.

Moreover, the number of times of appearances of k is the number of compositions of k into powers of 2, i.e., it is A023359(k), k>0. (End)

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).

Unsorted prime signature (A124010) is the sequence of exponents in a number's prime factorization.

All terms are normal (A055932), meaning their prime indices cover an initial interval of positive integers.

Unsorted prime signature is the sequence of exponents in a number's prime factorization.

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.

Unsorted prime signature (A124010) is the sequence of exponents in a number's prime factorization.

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.

For c-equivalence, see the comments to A233249. Briefly put, two positive integers m and n are c-equivalent in the sense of Vladimir Shevelev, if they have ordinary binary representations with the same multisets of substrings resulting from cutting the full strings immediately before each bit 1. A(n) is defined as the smallest positive integer, which is c-equivalent to n. Alternatively, in the manners of A114994, the lengths of these substrings can be considered as representing ways to write integers as sums of positive integers with arbitrarily ordered sums, and a(n) as the unique integer whose corresponding substring lengths form the corresponding integer partition.

For instance, the ordinary binary representations of 11, 13, and 14 are 1011, 1101, and 1110, respectively, which yields the equal multisets {"10","1","1"}, and {"1","10","1"}, and {"1","1","10"} of strings, respectively; whence 11, 13, and 14 are c-equivalent.

A(n) is an odd number if and only if the substring "1" appears at least once in the multiset. Since this is the case if and only if it also holds for the binary representation of n concatenated with itself, and A233312(n) = a(m) for the number m whose binary representation is this concatenation, we have a(n) == A233312(n) (mod 2) for all n. Moreover, empirical data has suggested that perhaps A233312(n)+1 == A171791(n+1) (mod 2) for all n >= 1. This relation holds in general if and only if a(n)+1 == A171791(n+1) for the same n, which in its turn is true if and only if the relation with fibbinary numbers first empirically observed by Paul D. Hanna in the comments to A171791 holds in general.

The sequence A163382 also maps n to a c-equivalent integer <=n; however, here, only cyclic permutations of the sequences of substrings are allowed. Thus, a more restricted equivalence relation is used; whence a(n) <= A163382(n) for all n. Equality holds for infinitely many n, including n = 1..37.