

A161664


a(n) = Sum_{i=1..n} (i  d(i)), where d(n) is the number of divisors of n (A000005).


9



0, 0, 1, 2, 5, 7, 12, 16, 22, 28, 37, 43, 54, 64, 75, 86, 101, 113, 130, 144, 161, 179, 200, 216, 238, 260, 283, 305, 332, 354, 383, 409, 438, 468, 499, 526, 561, 595, 630, 662, 701, 735, 776, 814, 853, 895, 940, 978, 1024, 1068, 1115, 1161, 1212, 1258, 1309
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OFFSET

1,4


COMMENTS

The original definition was: Safe periods for the emergence of cicada species on prime number cycles.
See Table 9 in reference, page 75, which together with the chart on page 73 (see link) provide a mathematical basis for the emergence of cicada species on prime number cycles.
Also the number of 2element nondividing subsets of {1, ..., n}. The a(6)=7 subsets of {1,2,3,4,5,6} with two elements where no element divides the other are: {2,3}, {2,5}, {3,4}, {3,5}, {4,5}, {4,6}, {5,6}.  Alois P. Heinz, Mar 08 2011
Sum of the number of proper nondivisors of all positive integers <= n.  Omar E. Pol, Feb 13 2014


REFERENCES

E. Haga, Eratosthenes goes bugs! Exploring Prime Numbers, 2007, pp 7180; first publication 1994.


LINKS



FORMULA



EXAMPLE

Referring to the chart referenced, when nth year = 7 there are 16 xmarkers.
These represent unsafe periods for cicada emergence: 2816=12 safe periods.
The percent of safe periods for the entire 7 years is 12/28=~42.86%; for year 7 alone the calculation is 5/7 = 71.43%, a relatively good time to emerge.


MAPLE

# second Maple program:
a:= proc(n) option remember; `if`(n<1, 0,
a(n1)+nnumtheory[tau](n))
end:


MATHEMATICA

a[n_] := n*(n+1)/2  Sum[ DivisorSigma[0, k], {k, n}]; Table[a[n], {n, 55}] (* JeanFrançois Alcover, Nov 07 2011 *)


PROG

(Python)
from math import isqrt
def A161664(n): return (lambda m: n*(n+1)//2+m*m2*sum(n//k for k in range(1, m+1)))(isqrt(n)) # Chai Wah Wu, Oct 08 2021


CROSSREFS



KEYWORD

easy,nonn


AUTHOR



EXTENSIONS

Simplified definition, offset corrected and partially edited by Omar E. Pol, Jun 18 2009


STATUS

approved



