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Numbers n for new peaks of floor(sigma(n)/primepi(n)).
1

%I #28 Jan 12 2018 14:08:49

%S 2,6,12,24,30,36,60,96,120,180,240,360,600,720,840,1080,1260,1680,

%T 2520,5040,7560,10080,12600,15120,20160,25200,27720,45360,50400,55440,

%U 83160

%N Numbers n for new peaks of floor(sigma(n)/primepi(n)).

%C sigma(n) = A000203(n), primepi(n) = A000720(n).

%C The sequence entries frequently are members of A002182 (highly composite numbers). Similar sequences can be generated by varying the "k" seen in the PARI code, for example to k=2.

%C Subsequence of A002093 (highly abundant numbers). - _Jens Kruse Andersen_, Jul 15 2014

%H Jens Kruse Andersen, <a href="/A244043/b244043.txt">Table of n, a(n) for n = 1..82</a>

%F Define A(n) = floor(A000203(n)/A000720(n)) for n >= 2. Then a(1) = 2 and for n >= 2 a(n) is the least k > a(n-1) such that A(k) > A(a(n-1)). - _Wolfdieter Lang_, Jul 03 2014

%e Example at n=2 (start), sigma(2)=3, primepi(2)=1 so the initial peak is 3.

%e We see a new peak (4) at n=6 from floor(12/3), a(2)=6.

%e We see new peak (5) at n=12 from floor(28/5), a(3)=12. No entry is defined for n<2.

%t Reap[For[peak = 0; n = 2, n < 10^5, n++, f = Floor[DivisorSigma[1, n] / PrimePi[n]]; If[f > peak, peak = f; Sow[n]]]][[2, 1]] (* _Jean-François Alcover_, Jan 12 2018 *)

%o (PARI) genit={my(maxx=100000);peak=3;k=1;n=3;optr=2;sptr=1;

%o write("A244043.csv",sptr," , ",2);while(n<maxx, a=primepi(n);b=sigma(n);

%o c=floor(b/a*1./k);optr++; if(c>peak,sptr++;peak=c;

%o write("A244043.csv",sptr," , ",optr););n++);}

%Y Cf. A000203, A000720, A002093, A002182.

%K easy,nonn

%O 1,1

%A _Bill McEachen_, Jun 17 2014

%E Edited. Crossrefs for sigma and primepi added. - _Wolfdieter Lang_, Jul 03 2014