A387383 - OEIS

A387383

Numbers k such that the cumulative sums of the k-th composition in graded reverse lexicographic order ("standard order", see A066099) do not cover all residue classes modulo p for any prime p; i.e., such that (0, x_1, ..., x_1+...+x_j) is an admissible prime tuple pattern, where (x_1, ..., x_j) is the k-th composition.

0, 2, 8, 32, 34, 40, 128, 130, 160, 162, 512, 520, 544, 552, 2048, 2050, 2056, 2080, 2082, 2088, 2176, 2178, 2208, 2210, 2560, 2568, 2592, 2600, 8192, 8194, 8224, 8226, 8320, 8322, 8352, 10240, 10242, 10272, 10274, 10368, 10370, 32768, 32776, 32800, 32808, 33280

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OFFSET

1,2

COMMENTS

If k is not a term, there are only finitely many primes p such that p, p+x_1, ..., p+x_1+...+x_j are all prime, where (x_1,...,x_j) is the k-th composition. If the first Hardy-Littlewood conjecture (or the k-tuple conjecture) is true, the converse also holds and this sequence is a subsequence of A387382.

There are A023189(j+1) terms with 2*j binary digits (with the convention that 0 has 0 digits), corresponding to the admissible prime patterns of diameter 2*j. There are no terms with an odd number of binary digits.

The number of terms less than 2^j is A292225(j+1).

The number of terms x less than 2^j with A000120(x) = i is A292224(j+1,i+1).

A161781 gives a similar representation of admissible prime tuple patterns (see formula).

LINKS

Pontus von Brömssen, Table of n, a(n) for n = 1..10000

Wikipedia, First Hardy-Littlewood conjecture.

Wikipedia, Prime k-tuple.

FORMULA

a(n) = (A000695(A161781(n-1))-1)/2.

EXAMPLE

1 is not a term, because the cumulative sums of the 1st composition (1) are (0,1), which cover both residue classes modulo 2.

2 is a term, because the cumulative sums of the 2nd composition (2) are (0,2), which do not cover all residue classes modulo any prime.