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

2, 6, 12, 24, 30, 36, 60, 96, 120, 180, 240, 360, 600, 720, 840, 1080, 1260, 1680, 2520, 5040, 7560, 10080, 12600, 15120, 20160, 25200, 27720, 45360, 50400, 55440, 83160, 110880, 138600, 166320, 196560, 221760, 277200

Offset: 1

Author

Bill McEachen, Jun 17 2014

Keywords

• easy
• nonn

Comments

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

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.

Subsequence of A002093 (highly abundant numbers). - Jens Kruse Andersen, Jul 15 2014

Examples

			Example at n=2 (start), sigma(2)=3, primepi(2)=1 so the initial peak is 3.
We see a new peak (4) at n=6 from floor(12/3), a(2)=6.
We see new peak (5) at n=12 from floor(28/5), a(3)=12. No entry is defined for n<2.
		

Links

• Jens Kruse Andersen, Table of n, a(n) for n = 1..82

Crossrefs

Cf. A000203, A000720, A002093, A002182.

Programs

Mathematica

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 *)
DeleteDuplicates[Table[{n,Floor[DivisorSigma[1,n]/PrimePi[n]]},{n,2,85000}],GreaterEqual[#1[[2]],#2[[2]]]&][[;;,1]] (* Harvey P. Dale, Mar 13 2025 *)

PARI

genit={my(maxx=100000);peak=3;k=1;n=3;optr=2;sptr=1;
write("A244043.csv",sptr," , ",2);while(npeak,sptr++;peak=c;
write("A244043.csv",sptr," , ",optr););n++);}

Formula

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

Extensions

Edited. Crossrefs for sigma and primepi added. - Wolfdieter Lang, Jul 03 2014

More terms from Harvey P. Dale, Mar 13 2025