A208767 Generalized 2-super abundant numbers.

1, 2, 4, 6, 12, 24, 48, 60, 120, 240, 360, 720, 840, 1680, 2520, 5040, 10080, 15120, 25200, 27720, 55440, 110880, 166320, 277200, 332640, 360360, 720720, 1441440, 2162160, 3603600, 4324320, 7207200, 10810800, 12252240, 21621600, 24504480, 36756720, 61261200

Offset: 1

Author

Ben Branman, Mar 01 2012

Keywords

• nonn

Comments

The generalized k-super abundant numbers are those such that sigma_k(n)/(n^k) > sigma_k(m)/(m^k) for all m < n, where sigma_k(n) is the sum of the k-th powers of the divisors of n.

1-super abundant numbers are A004394. 0-super abundant numbers are A002182.

Pillai called these numbers "highly abundant numbers of the 2nd order". - Amiram Eldar, Jun 30 2019

Examples

			For i=24, sigma_2(24)/(24^2)=850/576=1.47569, a new record, thus 24 is in the sequence.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..251
• S. Sivasankaranarayana Pillai, Highly abundant numbers, Bulletin of the Calcutta Mathematical Society, Vol. 35, No. 1 (1943), pp. 141-156.
• S. Sivasankaranarayana Pillai, On numbers analogous to highly composite numbers of Ramanujan, Rajah Sir Annamalai Chettiar Commemoration Volume, ed. Dr. B. V. Narayanaswamy Naidu, Annamalai University, 1941, pp. 697-704.
• Srinivasa Ramanujan, Highly composite numbers, Annotated and with a foreword by Jean-Louis Nicolas and Guy Robin, The Ramanujan Journal, Vol. 1, No. 2 (1997), pp. 119-153, alternative link.

Crossrefs

Cf. A002182, A004394, A004490, A002201, A001157, A013661, A017667, A017668.

Subsequence of A025487.

Programs

Mathematica

s = {1}; a = 1; Do[ If[DivisorSigma[2, n]/(n^2) > a, a = DivisorSigma[2, n]/(n^2); AppendTo[s, n]], {n, 10000000}]; s

Formula

Limit_{n->oo} A001157(a(n))/a(n)^2 = zeta(2) (A013661). - Amiram Eldar, Sep 25 2022

Extensions

More terms from Amiram Eldar, May 12 2019