A161664 - OEIS (original) (raw)

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

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 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.

FORMULA

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

# Alternative:

a:= proc(n) option remember; `if`(n<1, 0,

a(n-1)+n-numtheory[tau](n))

end:

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

EXTENSIONS

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