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A244049 Sum of all proper divisors of all positive integers <= n. 2
0, 0, 0, 2, 2, 7, 7, 13, 16, 23, 23, 38, 38, 47, 55, 69, 69, 89, 89, 110, 120, 133, 133, 168, 173, 188, 200, 227, 227, 268, 268, 298, 312, 331, 343, 397, 397, 418, 434, 483, 483, 536, 536, 575, 607, 632, 632, 707, 714, 756, 776, 821, 821, 886, 902 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

The proper divisors of n are all divisors except 1 and n itself. Therefore noncomposite numbers have no proper divisors.

For the sum of all aliquot divisors of all positive integers <= n see A153485.

For the sum all divisors of all positive integers <= n see A024916.

Partial sums of A048050.

LINKS

Table of n, a(n) for n=1..55.

FORMULA

a(n) = A024916(n) - A034856(n) = A153485(n) - n + 1.

a(n) = a(n - 1) if and only if n is prime.

G.f.: (1/(1 - x))*Sum_{k>=2} k*x^(2*k)/(1 - x^k). - Ilya Gutkovskiy, Jan 22 2017

EXAMPLE

a(4) = 2 because the only proper divisor of 4 is 2 and the previous n contributed no proper divisors to the sum.

a(5) = 2 because 5 is prime and contributes no proper divisors to the sum.

a(6) = 7 because the proper divisors of 6 are 2 and 3, which add up to 5, and a(5) + 5 = 2 + 5 = 7.

MATHEMATICA

propDivsRunSum[1] := 0; propDivsRunSum[n_] := propDivsRunSum[n] = propDivsRunSum[n - 1] + (Plus@@Divisors[n]) - (n + 1); Table[propDivsRunSum[n], {n, 60}] (* Alonso del Arte, Jun 30 2014 *)

Accumulate[Join[{0}, Table[Total[Most[Divisors[n]]]-1, {n, 2, 60}]]] (* Harvey P. Dale, Aug 12 2016 *)

CROSSREFS

Cf. A000040, A000203, A001065, A008578, A024916, A027750, A048050, A153485.

Sequence in context: A045923 A306238 A318086 * A271229 A199886 A117779

Adjacent sequences:  A244046 A244047 A244048 * A244050 A244051 A244052

KEYWORD

nonn,easy

AUTHOR

Omar E. Pol, Jun 24 2014

STATUS

approved

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Last modified September 19 12:57 EDT 2019. Contains 327198 sequences. (Running on oeis4.)