A079667 a(n) = (1/2) * Sum_{d divides n} abs(n/d-d).

0, 1, 2, 3, 4, 6, 6, 9, 8, 12, 10, 16, 12, 18, 16, 21, 16, 27, 18, 28, 24, 30, 22, 40, 24, 36, 32, 42, 28, 50, 30, 49, 40, 48, 36, 65, 36, 54, 48, 66, 40, 72, 42, 70, 60, 66, 46, 92, 48, 77, 64, 84, 52, 96, 60, 92, 72, 84, 58, 126, 60, 90, 82, 105, 72, 120, 66, 112, 88, 114, 70

Offset: 1

Author

Vladeta Jovovic, Jan 25 2003

Keywords

• easy
• nonn

Comments

Also, Sum_{i|n, sqrt(n)

References

• H. J. S. Smith, Report on the Theory of Numbers, reprinted in Vol. 1 of his Collected Math. Papers, Chelsea, NY, 1979, see p. 323.

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A060866, A116589.

Programs

Mathematica

Table[DivisorSum[n, Abs[n/# - #] &, # <= Sqrt[n] &], {n, 71}] (* Michael De Vlieger, Mar 17 2021 *)

PARI

a(n)=if(n<2, 0, sumdiv(n,d, abs(n/d-d))/2) /* Michael Somos, Nov 19 2005 */

SageMath

def A079667(n): return sum(n//d - d for d in divisors(n) if d*d <= n)
print([A079667(n) for n in range(1, 72)])  # Peter Luschny, Jan 01 2024

Formula

a(n) = A070038(n) - A066839(n).

G.f.: Sum_{k>0} x^(k^2+k)/(1-x^k)^2 . - Michael Somos, Nov 19 2005

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