cp's OEIS Frontend

This is a companion sequence to A295355. The sequence (without exact first and last terms as well as the number of terms) was found by Bays and Hudson in 1978 (see references). The full sequence up to 10^15 contains 6 sign-changing zones with 2381904 terms in total with A(2381904) = 699914738212849 as the last one.

We found the 7th sign-changing zone between 10^15 and 10^16. It starts with A(2381905) = 8744052767229817, ends with A(2792591) = 8772206355445549 and contains 410687 terms. - Andrey S. Shchebetov and Sergei D. Shchebetov, Apr 26 2019

This is the binary "early-birdness" of n (cf. A116700, A296364).

Theorem: a(n) > 0 for all n > 1.

Proof. The claim is true for 2 <= n <= 7, so assume n >= 8, and let u = 1... denote the binary expansion of n. Let L denote the list of all binary vectors whose concatenation gives A076478.

To show a(n)>0 it is enough to exhibit a pair of successive binary vectors b, c in L whose concatenation contains a copy of u that begins in b and is such that b appears in L before u does. There are three cases.

(i) Suppose n is even, say u = 1x0. Take c = x00, and let b be the vector preceding c in L, so that b = y11, say. Then bc = y11x00 contains u.

(ii) Suppose n = 2^k-1, u = 1^k. Take b = 01^(k-1), c = 10^(k-1), so that bc = 0 1^k 0^(k-1).

(iii) Otherwise, n is an odd number whose binary expansion contains a 0, say u = 1^k 0x1. Take c = 0x10^k, and let b be the vector preceding c in L, so that b = y1^k, say, and bc = y1^k 0x10^k.

In each case we need to verify that b does appear in L before u, but we leave this easy verification to the reader. QED

This is a companion sequence to A297450 and the first discovered for pi_{24,17}(p) - pi_{24,1}(p) prime race. The full sequence up to 10^15 contains 3 sign-changing zones with 963922 terms in total with A(963922) = 23241097440243 as the last one.

This is a companion sequence to A297449 and the first discovered for pi_{24,17}(p) - pi_{24,1}(p) prime race. The full sequence up to 10^15 contains 3 sign-changing zones with 963922 terms in total with A(963922) = 772739867710897 as the last one.

This is a companion sequence to A298821 and the first discovered for pi_{24,19}(p) - pi_{24,1}(p) prime race. The full sequence up to 10^15 contains 5 sign-changing zones with 3436990 terms in total with A(3436990) = 23049274819456 as the last one.

This is a companion sequence to A298820 and the first discovered for pi_{24,19}(p) - pi_{24,1}(p) prime race. The full sequence up to 10^15 contains 5 sign-changing zones with 3436990 terms in total with A(3436990) = 766164822666883 as the last one.