A373823 Half the sum of the n-th maximal run of first differences of odd primes.

2, 2, 1, 2, 1, 2, 3, 1, 3, 2, 1, 2, 6, 1, 3, 2, 1, 3, 2, 3, 4, 2, 1, 2, 1, 2, 7, 2, 3, 1, 5, 1, 6, 2, 6, 1, 5, 1, 2, 1, 12, 2, 1, 2, 3, 1, 5, 9, 1, 3, 2, 1, 5, 7, 2, 1, 2, 7, 3, 5, 1, 2, 3, 4, 6, 2, 3, 4, 2, 4, 5, 1, 5, 1, 3, 2, 3, 4, 2, 1, 2, 6, 4, 2, 4, 2, 3

Offset

1,1

Comments

Halved run-sums of A001223.

Links

Table of n, a(n) for n=1..87.

Example

The odd primes are:
3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, ...
with first differences:
2, 2, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 6, 2, 6, 4, 2, 6, 4, 6, 8, ...
with runs:
(2,2), (4), (2), (4), (2), (4), (6), (2), (6), (4), (2), (4), (6,6), ...
with halved sums a(n).

Mathematica

Total/@Split[Differences[Select[Range[3, 1000], PrimeQ]]]/2

Crossrefs

Halved run-sums of A001223.

For run-lengths we have A333254, run-lengths of run-lengths A373821.

Multiplying by two gives A373822.

A000040 lists the primes.

A027833 gives antirun lengths of odd primes (partial sums A029707).

A046933 counts composite numbers between primes.

A065855 counts composite numbers up to n.

A071148 gives partial sums of odd primes.

A373820 gives run-lengths of antirun-lengths of odd primes.

For prime runs: A001359, A006512, A025584, A067774, A373405, A373406.

Cf. A037201, A038664, A054265, A176246, A251092, A373404, A373825.

Sequence in context: A029424 A373121 A061498 * A106029 A260311 A188431

Adjacent sequences: A373820 A373821 A373822 * A373824 A373825 A373826

Keyword

nonn

Author

Gus Wiseman, Jun 22 2024

Status

approved