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

Author

Michael De Vlieger, Dec 03 2024

Keywords

• nonn
• easy

Examples

			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.
		

Links

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

Crossrefs

Cf. A001223, A061398, A080101, A126706.

Programs

Mathematica

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

Formula

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