A248517 Number of odd divisors > 1 in the numbers 1 through n, counted with multiplicity.

0, 0, 0, 1, 1, 2, 3, 4, 4, 6, 7, 8, 9, 10, 11, 14, 14, 15, 17, 18, 19, 22, 23, 24, 25, 27, 28, 31, 32, 33, 36, 37, 37, 40, 41, 44, 46, 47, 48, 51, 52, 53, 56, 57, 58, 63, 64, 65, 66, 68, 70, 73, 74, 75, 78, 81, 82, 85, 86, 87, 90, 91, 92, 97, 97, 100, 103, 104, 105, 108, 111

Offset

0,6

Comments

Number of partitions of n into 3 parts such that the smallest part divides the "middle" part. - Wesley Ivan Hurt, Oct 21 2021

Links

Charles R Greathouse IV, Table of n, a(n) for n = 0..10000

Formula

a(n) = Sum_{j=1..n} A069283(j).

a(n) = A060831(n) - n.

a(n) = A006218(n) - A006218(floor(n/2)) - n. - Charles R Greathouse IV, Jun 18 2015

a(n) = Sum_{i=1..n-1} floor(floor(i/2)/(n-i)). - Wesley Ivan Hurt, Jan 30 2016

Maple

A248517 := proc(n)
    add(A069283(j), j=1..n) ;
end proc:

Mathematica

Table[Sum[Floor[Floor[i/2]/(n - i)], {i, n - 1}], {n, 0, 100}] (* Wesley Ivan Hurt, Jan 30 2016 *)
Join[{0}, Accumulate[Table[Count[Divisors[n], _?OddQ]-1, {n, 80}]]] (* Harvey P. Dale, Jan 06 2019 *)
Join[{0}, Accumulate[Table[DivisorSigma[0, n/2^IntegerExponent[n, 2]] - 1, {n, 1, 100}]]] (* Amiram Eldar, Jul 10 2022 *)

Prog

(PARI) a(n)=my(n2=n\2); sum(k=1, sqrtint(n), n\k)*2-sqrtint(n)^2-sum(k=1, sqrtint(n2), n2\k)*2+sqrtint(n2)^2-n \\ Charles R Greathouse IV, Jun 18 2015
(Python)
from math import isqrt
def A248517(n): return ((t:=isqrt(m:=n>>1))+(s:=isqrt(n)))*(t-s)+(sum(n//k for k in range(1, s+1))-sum(m//k for k in range(1, t+1))<<1)-n # Chai Wah Wu, Oct 23 2023

Crossrefs

Cf. A002541, A006218, A069283.

Sequence in context: A350656 A100476 A290083 * A364031 A007896 A351006

Adjacent sequences: A248514 A248515 A248516 * A248518 A248519 A248520

Keyword

nonn,easy

Author

R. J. Mathar, Jun 18 2015

Status

approved