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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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,4
COMMENTS
Partial Sums of A049820. - Omar E. Pol, Jun 18 2009
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 2-element 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 71-80; first publication 1994.
LINKS
G. F. Webb, The prime number periodical Cicada problem, Discr. Cont. Dyn. Syst. 1 (3) (2001) 387
Wildforests, Cicada, visited Dec. 2012. - From N. J. A. Sloane, Dec 25 2012
FORMULA
a(n) = A000217(n) - A006218(n).
EXAMPLE
a(8) in A000217 minus a(8) in A006218 = a(7) above (28-16=12).
Referring to the chart referenced, when n-th year = 7 there are 16 x-markers.
These represent unsafe periods for cicada emergence: 28-16=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
with(numtheory): A161664:=n->add(i-tau(i), i=1..n): seq(A161664(n), n=1..100); # Wesley Ivan Hurt, Jul 15 2014
# second Maple program:
a:= proc(n) option remember; `if`(n<1, 0,
a(n-1)+n-numtheory[tau](n))
end:
seq(a(n), n=1..55); # Alois P. Heinz, Jun 24 2022
MATHEMATICA
a[n_] := n*(n+1)/2 - Sum[ DivisorSigma[0, k], {k, n}]; Table[a[n], {n, 55}] (* Jean-François Alcover, Nov 07 2011 *)
PROG
(Python)
from math import isqrt
def A161664(n): return (lambda m: n*(n+1)//2+m*m-2*sum(n//k for k in range(1, m+1)))(isqrt(n)) # Chai Wah Wu, Oct 08 2021
CROSSREFS
Column 2 of triangle A187489 or of A355145.
Sequence in context: A343944 A226084 A294861 * A080547 A080555 A320657
KEYWORD
easy,nonn
AUTHOR
Enoch Haga, Jun 15 2009
EXTENSIONS
Simplified definition, offset corrected and partially edited by Omar E. Pol, Jun 18 2009
New name from Wesley Ivan Hurt, Jul 15 2014
STATUS
approved

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Last modified May 24 21:45 EDT 2024. Contains 372782 sequences. (Running on oeis4.)