A161664 a(n) = Sum_{i=1..n} (i - d(i)), where d(n) is the number of divisors of n (A000005).

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

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

Enoch Haga, Eratosthenes goes bugs! Exploring Prime Numbers, 2007, pp 71-80; first publication 1994.

Links

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

A. Baker, Are there Genuine Mathematical Explanations of Physical Phenomena?, Mind 114 (454) (2005) 223-238.

Enoch Haga, Prime Safe Periods

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).

For n>1: a(n) = Sum_{h=1..n} Sum_{m=1..1 + 2*floor(n/2 - 1/2)} Sum_{k=1 + floor(h/(m + 1))..floor(h/m - 1/m)} 1 (from Granvik at A368592). - Bill McEachen, Apr 01 2025

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
# Alternative:
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

Cf. A000005, A000217, A049820, A006218, A051014.

Column 2 of triangle A187489 or of A355145.

Sequence in context: A343944 A226084 A294861 * A080547 A080555 A320657

Adjacent sequences: A161661 A161662 A161663 * A161665 A161666 A161667

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