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A244043 Numbers n for new peaks of floor(sigma(n)/primepi(n)).

%I A244043 #32 Mar 13 2025 16:37:29
%S A244043 2,6,12,24,30,36,60,96,120,180,240,360,600,720,840,1080,1260,1680,
%T A244043 2520,5040,7560,10080,12600,15120,20160,25200,27720,45360,50400,55440,
%U A244043 83160,110880,138600,166320,196560,221760,277200
%N A244043 Numbers n for new peaks of floor(sigma(n)/primepi(n)).
%C A244043 sigma(n) = A000203(n), primepi(n) = A000720(n).
%C A244043 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 A244043 Subsequence of A002093 (highly abundant numbers). - _Jens Kruse Andersen_, Jul 15 2014
%H A244043 Jens Kruse Andersen, <a href="/A244043/b244043.txt">Table of n, a(n) for n = 1..82</a>
%F A244043 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 A244043 Example at n=2 (start), sigma(2)=3, primepi(2)=1 so the initial peak is 3.
%e A244043 We see a new peak (4) at n=6 from floor(12/3), a(2)=6.
%e A244043 We see new peak (5) at n=12 from floor(28/5), a(3)=12. No entry is defined for n<2.
%t A244043 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 *)
%t A244043 DeleteDuplicates[Table[{n,Floor[DivisorSigma[1,n]/PrimePi[n]]},{n,2,85000}],GreaterEqual[#1[[2]],#2[[2]]]&][[;;,1]] (* _Harvey P. Dale_, Mar 13 2025 *)
%o A244043 (PARI) genit={my(maxx=100000);peak=3;k=1;n=3;optr=2;sptr=1;
%o A244043 write("A244043.csv",sptr," , ",2);while(n<maxx, a=primepi(n);b=sigma(n);
%o A244043 c=floor(b/a*1./k);optr++; if(c>peak,sptr++;peak=c;
%o A244043 write("A244043.csv",sptr," , ",optr););n++);}
%Y A244043 Cf. A000203, A000720, A002093, A002182.
%K A244043 easy,nonn
%O A244043 1,1
%A A244043 _Bill McEachen_, Jun 17 2014
%E A244043 Edited. Crossrefs for sigma and primepi added. - _Wolfdieter Lang_, Jul 03 2014
%E A244043 More terms from _Harvey P. Dale_, Mar 13 2025