login
This site is supported by donations to The OEIS Foundation.

 

Logo

The October issue of the Notices of the Amer. Math. Soc. has an article about the OEIS.

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A161664 Sum_{i=1..n} i-d(i), where d(n) is the number of divisors of n (A000005). 6

%I

%S 0,0,1,2,5,7,12,16,22,28,37,43,54,64,75,86,101,113,130,144,161,179,

%T 200,216,238,260,283,305,332,354,383,409,438,468,499,526,561,595,630,

%U 662,701,735,776,814,853,895,940,978,1024,1068,1115,1161,1212,1258,1309

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

%C Partial Sums of A049820 - _Omar E. Pol_, Jun 18 2009.

%C The original definition was: Safe periods for the emergence of cicada species on prime number cycles.

%C 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.

%C 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

%C Sum of the number of proper nondivisors of all positive integers <= n. - _Omar E. Pol_, Feb 13 2014

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

%H Alois P. Heinz, <a href="/A161664/b161664.txt">Table of n, a(n) for n = 1..1000</a>

%H A. Baker, <a href="http://dx.doi.org/10.1093/mind/fzi223">Are there Genuine Mathematical Explanations of Physical Phenomena?</a>, Mind 114 (454) (2005) 223-238.

%H E. Haga, <a href="/A161664/a161664.pdf">Prime Safe Periods</a>

%H G. F. Webb, <a href="http://philoscience.unibe.ch/lehre/winter06/wtwg_bio/webb01.pdf">The prime number periodical Cicada problem</a>, Discr. Cont. Dyn. Syst. 1 (3) (2001) 387

%H Wildforests, <a href="http://wiki.wildforests.co/topic/Cicada">Cicada</a>, visited Dec. 2012. - From _N. J. A. Sloane_, Dec 25 2012

%F a(n) = A000217(n) - A006218(n).

%e a(8) in A000217 minus a(8) in A006218 = a(7) above (28-16=12).

%e Referring to the chart referenced, when n-th year = 7 there are 16 x-markers.

%e These represent unsafe periods for cicada emergence: 28-16=12 safe periods.

%e 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.

%p with(numtheory): A161664:=n->add(i-tau(i), i=1..n): seq(A161664(n), n=1..100); # _Wesley Ivan Hurt_, Jul 15 2014

%t a[n_] := n*(n+1)/2 - Sum[ DivisorSigma[0, k], {k, n}]; Table[a[n], {n, 55}] (* _Jean-Fran├žois Alcover_, Nov 07 2011 *)

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

%Y Column 2 of triangle A187489.

%K easy,nonn

%O 1,4

%A _Enoch Haga_, Jun 15 2009

%E Simplified definition, offset corrected and partially edited by _Omar E. Pol_, Jun 18 2009

%E New name from _Wesley Ivan Hurt_, Jul 15 2014

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified September 22 17:05 EDT 2018. Contains 315270 sequences. (Running on oeis4.)