A337328 Number of pairs of squarefree divisors of n, (d1,d2), such that d1 <= d2.

1, 3, 3, 3, 3, 10, 3, 3, 3, 10, 3, 10, 3, 10, 10, 3, 3, 10, 3, 10, 10, 10, 3, 10, 3, 10, 3, 10, 3, 36, 3, 3, 10, 10, 10, 10, 3, 10, 10, 10, 3, 36, 3, 10, 10, 10, 3, 10, 3, 10, 10, 10, 3, 10, 10, 10, 10, 10, 3, 36, 3, 10, 10, 3, 10, 36, 3, 10, 10, 36, 3, 10, 3, 10, 10, 10, 10, 36

Offset

1,2

Links

Table of n, a(n) for n=1..78.

Formula

a(n) = Sum_{d1|n, d2|n, d1 and d2 squarefree, d1 <= d2} 1.

Example

a(28) = 10; The divisors of 28 are {1,2,4,7,14,28}. The pairs of squarefree divisors of 28, (d1,d2), such that d1 <= d2 are (1,1), (1,2), (1,7), (1,14), (2,2), (2,7), (2,14), (7,7), (7,14) and (14,14). So a(28) = 10.
a(29) = 3; The divisors of 29 are {1,29}. The pairs of squarefree divisors of 29, (d1,d2), such that d1 <= d2 are (1,1), (1,29) and (29,29). So a(29) = 3.
a(30) = 36; The divisors of 30 are {1,2,3,5,6,10,15,30}. The 36 pairs of squarefree divisors of 30, (d1,d2) such that d1 <= d2 are (1,1), (1,2), (1,3), (1,5), (1,6), (1,10), (1,15), (1,30), (2,2), (2,3), (2,5), (2,6), (2,10), (2,15), (2,30), (3,3), (3,5), (3,6), (3,10), (3,15), (3,30), (5,5), (5,6), (5,10), (5,15), (5,30), (6,6), (6,10), (6,15), (6,30), (10,10), (10,15), (10,30), (15,15), (15,30), (30,30). So a(30) = 36.
a(31) = 3; The divisors of 31 are {1,31}. The pairs of squarefree divisors of 31 are (1,1), (1,31) and (31,31). So a(31) = 3.

Mathematica

Table[Sum[Sum[MoebiusMu[i]^2 MoebiusMu[k]^2*(1 - Ceiling[n/k] + Floor[n/k]) (1 - Ceiling[n/i] + Floor[n/i]), {i, k}], {k, n}], {n, 100}]

Crossrefs

Sequence in context: A111575 A161836 A087576 * A201700 A087575 A001270

Adjacent sequences: A337325 A337326 A337327 * A337329 A337330 A337331

Keyword

nonn

Author

Wesley Ivan Hurt, Aug 23 2020

Status

approved