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A161664 Sum_{i=1..n} i-d(i), where d(n) is the number of divisors of n (A000005). 6


%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

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