A349758 - OEIS

A349758

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

%I #13 Dec 06 2021 03:12:52

%S 60,84,90,96,108,126,132,140,150,156,160,180,198,204,220,224,228,234,

%T 240,252,260,276,294,300,306,308,315,336,340,342,348,350,352,360,364,

%U 372,380,396,414,416,420,432,444,460,476,480,486,490,492,495,500,504,516

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

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

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

%D 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.

%H Amiram Eldar, <a href="/A349758/b349758.txt">Table of n, a(n) for n = 1..10000</a>

%H Jason Earls, <a href="https://dl.acm.org/doi/abs/10.5555/1006498.1006546">Some Smarandache-type sequences and problems concerning abundant and deficient numbers</a>, Smarandache Notions Journal, Vol. 14, No. 1 (2004), pp. 243-250.

%H József Sándor, <a href="https://blngcc.files.wordpress.com/2008/11/jozsel-sandor-selected-chaters-of-geometry-analysis-and-number-theory.pdf">Selected Chapters of Geometry, Analysis and Number Theory</a>, 2005, pp. 132-134.

%H Shikha Yadav and Surendra Yadav, <a href="http://raops.org.in/epapers/dec16_10.pdf">Multiplicatively perfect and related numbers</a>, Journal of Rajasthan Academy of Physical Sciences, Vol. 15, No. 4 (2016), pp. 345-350.

%e 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.

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

%o (PARI) isab(k) = sigma(k) > 2*k; \\ A005101

%o isok(k) = my(f=factor(k)); isab(numdiv(f)) && isab(sigma(f)); \\ _Michel Marcus_, Dec 02 2021

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

%Y A349760 is a subsequence.

%K nonn

%O 1,1

%A _Amiram Eldar_, Nov 29 2021