cp's OEIS Frontend

It appears that a(n) gives the position of its own n-th 1 modulo 3 term, the n-th 2 modulo 3 term in A026602, and the n-th multiple of 3 in A026603. A026602 and A026603 appear to have analogous indexical properties. - Matthew Vandermast, Oct 06 2010

This follows directly from the generating morphism for A026600: a 1 in position k creates a 1 in position 3k-2, a 2 in position 3k-1, and a 3 in position 3k. Since each block of three terms in A026600 is a permutation of {1,2,3}, these created terms are the k-th terms of their respective index sequences. The proof for the other index sequences is similar. - Charlie Neder, Mar 10 2019

It appears that Sum[k^j, 0<=k<=2^n-1, k in A198680] = Sum[k^j, 0<=k<=2^n-1, k in A198681] = Sum[k^j, 0<=k<=2^n-1, k in A180682], for 0<=j<=n-1, which has been verified numerically in a number of cases. This is a generalization of Prouhet's Theorem (see the reference). To illustrate for j=3, we have Sum[k^3, 0<=k<=2^n-1, k in A198680] = {0, 0, 12636, 1108809, 94478400, 7780827681, 633724260624, 51425722195929, 4168024588857600,...}, Sum[k^3, 0<=k<=2^n-1, k in A198681] = {0, 27, 14580, 1095687, 94478400, 7780827681, 633724260624, 51425722195929, 4168024588857600,..., Sum[k^3, 0<=k<=2^n-1, k in A198682] = {0, 216, 7776, 1121931, 94478400, 7780827681, 633724260624, 51425722195929, 4168024588857600,...}, and it is seen that all three sums agree for n>=4=j+1.

This sequence is infinite. Given any arbitrarily large number, the last 4 binary bits can be set to 0, and, if this number does not already meet the criteria, one of the last 4 bits can be increased to 1 such that it does.