A058267 -id:A058267 - cp's OEIS frontend

A332931 Sum of round(sqrt(d)) where d runs through the divisors of n.

1, 2, 3, 4, 3, 6, 4, 7, 6, 7, 4, 11, 5, 9, 9, 11, 5, 13, 5, 13, 11, 10, 6, 19, 8, 11, 11, 16, 6, 20, 7, 17, 12, 12, 12, 24, 7, 12, 13, 22, 7, 24, 8, 19, 19, 14, 8, 30, 11, 19, 14, 20, 8, 25, 13, 26, 15, 15, 9, 37, 9, 16, 22, 25, 15, 28, 9, 22, 16, 28, 9, 40

Offset: 1

Harvey P. Dale, Mar 02 2020

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A058267 (which has the "round" outside the sum), A086671, A332932, A332933, A332934, A332935.

a:= n-> add(round(sqrt(d)), d=numtheory[divisors](n)):
seq(a(n), n=1..80);  # Alois P. Heinz, Mar 02 2020

Table[DivisorSum[n,Floor[1/2+Sqrt[#]]&],{n,80}]

a(n) = sumdiv(n, d, round(sqrt(d))); \\ Michel Marcus, Mar 03 2020

from math import isqrt
from sympy import divisors
def A332931(n): return sum((m:=isqrt(d))+int(d-m*(m+1)>=1) for d in divisors(n,generator=True)) # Chai Wah Wu, Aug 03 2022

A058266 An approximation to sigma_{1/2}(n): floor( sum_{ d divides n } sqrt(d) ).

Original entry on oeis.org

1, 2, 2, 4, 3, 6, 3, 7, 5, 7, 4, 12, 4, 8, 8, 11, 5, 13, 5, 14, 9, 10, 5, 19, 8, 11, 10, 16, 6, 21, 6, 16, 11, 12, 11, 25, 7, 12, 12, 23, 7, 24, 7, 19, 18, 13, 7, 30, 10, 19, 13, 20, 8, 26, 13, 26, 14, 15, 8, 39, 8, 15, 20, 24, 14, 28, 9, 22, 15, 28, 9, 41, 9

Offset: 1

N. J. A. Sloane, Dec 08 2000

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000203, A001157, A058267, A058268, A078434, A086671.

with(numtheory); f := proc(n) local d, t1, t2; t2 := 0; t1 := divisors(n); for d in t1 do t2 := t2 + sqrt(d) end do; t2 end proc; # exact value of sigma_{1/2}(n)
with(numtheory):seq(floor(sigma[1/2](n)),n=1..80);

f[n_] := Floor@DivisorSigma[1/2, n]; Array[f, 73] (* Robert G. Wilson v, Aug 17 2017*)

a(n) = floor(sumdiv(n, d, sqrt(d))); \\ Michel Marcus, Aug 17 2017

Sum_{k=1..n} a(k) ~ (2/3)*zeta(3/2) * n^(3/2). - Amiram Eldar, Jan 14 2023

A058268 An approximation to sigma_{1/2}(n): ceiling( sum_{d|n} sqrt(d) ).

Original entry on oeis.org

1, 3, 3, 5, 4, 7, 4, 8, 6, 8, 5, 13, 5, 9, 9, 12, 6, 14, 6, 15, 10, 11, 6, 20, 9, 12, 11, 17, 7, 22, 7, 17, 12, 13, 12, 26, 8, 13, 13, 24, 8, 25, 8, 20, 19, 14, 8, 31, 11, 20, 14, 21, 9, 27, 14, 27, 15, 16, 9, 40, 9, 16, 21, 25, 15, 29, 10, 23, 16, 29, 10, 42

Offset: 1

N. J. A. Sloane, Dec 08 2000

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000203, A001157, A058266, A058267, A078434.

with(numtheory); f := proc(n) local d, t1, t2; t2 := 0; t1 := divisors(n); for d in t1 do t2 := t2 + sqrt(d) end do; t2 end proc; # exact value of sigma_{1/2}(n)

a[n_] := Ceiling[DivisorSigma[1/2, n]]; Array[a, 70] (* Amiram Eldar, Jan 14 2023 *)

Sum_{k=1..n} a(k) ~ (2/3)*zeta(3/2) * n^(3/2). - Amiram Eldar, Jan 14 2023

cp's OEIS Frontend

A332931 Sum of round(sqrt(d)) where d runs through the divisors of n.

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A058266 An approximation to sigma_{1/2}(n): floor( sum_{ d divides n } sqrt(d) ).

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A058268 An approximation to sigma_{1/2}(n): ceiling( sum_{d|n} sqrt(d) ).

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