A373643 Number of k between consecutive primes such that k is neither squarefree nor prime powers.

0, 0, 0, 0, 1, 0, 1, 1, 2, 0, 1, 1, 0, 2, 3, 2, 1, 1, 1, 1, 2, 1, 2, 3, 3, 0, 1, 1, 1, 5, 0, 3, 0, 4, 1, 3, 2, 1, 3, 2, 1, 3, 1, 1, 1, 4, 3, 2, 1, 1, 2, 1, 5, 1, 2, 2, 1, 3, 2, 0, 3, 6, 1, 1, 2, 4, 3, 4, 1, 3, 1, 3, 3, 3, 1, 3, 2, 1, 3, 3, 1, 4, 1, 1, 2, 2, 3

Offset

1,9

Links

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Formula

a(n) = A001223(n) - A061398(n) - A080101(n) - 1.

Example

Let S = A126706, the sequence of k neither squarefree nor prime powers.
a(1..4) = 0 since S(1) = 12.
a(5) = 1 since (11, 12, 13) contains S(1) = 12.
a(6) = 0 since (13, 14, 15, 16, 17) contains no number in S.
a(7) = 1 since (17, 18, 19) contains S(2) = 18.
a(8) = 1 since (19, 20, 21, 22, 23) contains S(3) = 20.
a(9) = 2 since (23, 24, 25, 26, 27, 28, 29) contains S(4) = 24 and S(5) = 28, etc.

Mathematica

Table[Count[Range[Prime[i] + 1, Prime[i + 1] - 1], _?(Nor[SquareFreeQ[#], PrimePowerQ[#]] &)], {i, 120}]

Crossrefs

Cf. A001223, A061398, A080101, A126706.

Sequence in context: A234954 A321201 A180649 * A191238 A049310 A168561

Adjacent sequences: A373640 A373641 A373642 * A373644 A373645 A373646

Keyword

nonn,easy

Author

Michael De Vlieger, Dec 03 2024

Status

approved