A349758 Nobly abundant numbers: numbers k such that both d(k) = A000005(k) and sigma(k) = A000203(k) are abundant numbers (A005101).

60, 84, 90, 96, 108, 126, 132, 140, 150, 156, 160, 180, 198, 204, 220, 224, 228, 234, 240, 252, 260, 276, 294, 300, 306, 308, 315, 336, 340, 342, 348, 350, 352, 360, 364, 372, 380, 396, 414, 416, 420, 432, 444, 460, 476, 480, 486, 490, 492, 495, 500, 504, 516

Offset: 1

Author

Amiram Eldar, Nov 29 2021

Keywords

• nonn

Comments

Analogous to sublime numbers (A081357), with abundant numbers instead of perfect numbers.

The least odd term is a(27) = 315 and the least term that is coprime to 6 is a(298) = 1925.

Examples

			60 is a term since both d(60) = 12 and sigma(60) = 168 are abundant numbers: sigma(12) = 28 > 2*12 = 24 and sigma(168) = 480 > 2*168 = 336.
		

References

• József Sándor and E. Egri, Arithmetical functions in algebra, geometry and analysis, Advanced Studies in Contemporary Mathematics, Vol. 14, No. 2 (2007), pp. 163-213.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Jason Earls, Some Smarandache-type sequences and problems concerning abundant and deficient numbers, Smarandache Notions Journal, Vol. 14, No. 1 (2004), pp. 243-250.
• József Sándor, Selected Chapters of Geometry, Analysis and Number Theory, 2005, pp. 132-134.
• Shikha Yadav and Surendra Yadav, Multiplicatively perfect and related numbers, Journal of Rajasthan Academy of Physical Sciences, Vol. 15, No. 4 (2016), pp. 345-350.

Crossrefs

Cf. A000005, A000203, A005101, A081357, A349759.

A349760 is a subsequence.

Programs

Mathematica

abQ[n_] := DivisorSigma[1, n] > 2*n; nobAbQ[n_] := And @@ abQ /@ DivisorSigma[{0, 1}, n]; Select[Range[500], nobAbQ]

PARI

isab(k) = sigma(k) > 2*k; \\ A005101
isok(k) = my(f=factor(k)); isab(numdiv(f)) && isab(sigma(f)); \\ Michel Marcus, Dec 02 2021