A013928 -id:A013928 - cp's OEIS frontend

A368642 a(n) = Sum_{k=1..n} mu(k)^2 * ceiling(n/k), where mu is the Möbius function (A008683).

1, 3, 6, 8, 11, 14, 19, 21, 23, 26, 31, 33, 38, 41, 46, 50, 53, 55, 60, 62, 67, 72, 77, 79, 83, 86, 90, 92, 97, 100, 109, 111, 114, 119, 124, 128, 133, 136, 141, 145, 150, 153, 162, 164, 168, 173, 178, 180, 184, 186, 191, 195, 200, 202, 207, 211, 216, 221, 226, 228

Offset: 1

Wesley Ivan Hurt, Jan 01 2024

Cf. A008683 (mu), A008966, A013928, A034444, A064608.

Table[Sum[MoebiusMu[k]^2*Ceiling[n/k], {k, n}], {n, 100}]

a(n) = A013928(n) + A008966(n) + A064608(n) - A034444(n).

A379771 Number of k <= n that are neither squarefree nor prime powers.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 7, 7, 7, 7, 8, 9, 9, 9, 10, 10, 11, 11, 12, 12, 13, 13, 14, 14, 14, 14, 15, 15, 15, 16, 16, 16, 16, 16, 17, 17, 17, 17, 18, 18, 18, 19, 20, 20

Offset: 1

Michael De Vlieger, Jan 10 2025

Partial sums of A355447.

			a(n) = 0 for n = 1..11, since 12 is the smallest number that is neither squarefree nor a prime power.
a(n) = 1 for n = 12..17, since the only k <= n that is neither squarefree nor a prime power is 12.
a(n) = 2 for n = 18..19, since 12 and 18 are the only numbers in A126706 that do not exceed n.
a(n) = 3 for n = 20..23, since 12, 18, and 20 are the only numbers in A126706 that do not exceed n, etc.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^20.

Cf. A013928, A065515, A126706, A355447.

Array[Count[Range[#], _?(Nor[SquareFreeQ[#], PrimePowerQ[#]] &)] &, 120]

a(n) = sum(k=1, n, !issquarefree(k) && !isprimepower(k)); \\ Michel Marcus, Jan 11 2025

from math import isqrt
from sympy import mobius, integer_nthroot, primepi
def A379771(n): return -sum(mobius(k)*(n//k**2) for k in range(2,isqrt(n)+1))-sum(primepi(integer_nthroot(n,k)[0]) for k in range(2,n.bit_length())) # Chai Wah Wu, Jan 22 2025

A126706(n) = m = positions of 1's in A355447; a(m) - a(m-1) = 1.

A209628 Number of squarefree numbers < n that are not prime.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 4, 5, 5, 5, 5, 5, 5, 6, 7, 7, 7, 7, 8, 8, 8, 8, 9, 9, 9, 10, 11, 12, 12, 12, 13, 14, 14, 14, 15, 15, 15, 15, 16, 16, 16, 16, 16, 17, 17, 17, 17, 18, 18, 19, 20, 20, 20, 20, 21, 21, 21, 22, 23, 23, 23, 24, 25, 25, 25, 25, 26

Offset: 1

Enrique Pérez Herrero, Mar 13 2012

nonn

Cf. A000720, A013928, A005117, A000040.

Accumulate[Table[Boole[SquareFreeQ[n]&&!PrimeQ[n]],{n,1,100}]]

a(n) = A013928(n+1) - A000720(n). [Corrected by Georg Fischer, Dec 13 2022]

A295164 Number of squarefree numbers m <= n that have a prime divisor greater than sqrt(n) (i.e., A006530(m) > sqrt(n)).

Original entry on oeis.org

0, 1, 2, 1, 2, 3, 4, 4, 2, 3, 4, 4, 5, 6, 7, 7, 8, 8, 9, 9, 10, 11, 12, 12, 9, 10, 10, 10, 11, 11, 12, 12, 13, 14, 15, 15, 16, 17, 18, 18, 19, 20, 21, 21, 21, 22, 23, 23, 18, 18, 19, 19, 20, 20, 21, 21, 22, 23, 24, 24, 25, 26, 26, 26, 27, 28, 29, 29, 30, 30, 31, 31, 32, 33, 33, 33, 34, 35, 36, 36, 36, 37, 38, 38, 39, 40, 41, 41, 42, 42, 43, 43, 44, 45, 46, 46, 47, 47, 47, 47

Offset: 1

Max Alekseyev, Nov 16 2017

nonn

Cf. A241419.

a(n) = Sum_{prime p > sqrt(n)} A013928(floor(n/p)+1).

a(n) = A013928(n+1) - A295101(n).

cp's OEIS Frontend

A368642 a(n) = Sum_{k=1..n} mu(k)^2 * ceiling(n/k), where mu is the Möbius function (A008683).

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A379771 Number of k <= n that are neither squarefree nor prime powers.

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A209628 Number of squarefree numbers < n that are not prime.

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A295164 Number of squarefree numbers m <= n that have a prime divisor greater than sqrt(n) (i.e., A006530(m) > sqrt(n)).

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