This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A298973 #19 Jul 20 2020 01:55:01
%S A298973 70,1430,1870,2002,2090,2210,2470,2530,2990,3190,3230,3410,3770,4030,
%T A298973 4070,4510,4730,5170,5830,15015,19635,21945,23205,25935,26565,31395,
%U A298973 33495,33915,35805,39585,41055,42315,42735,45885,47355,49665,49742,50505,51765,54285,55965,58695,58786,60214,61215,64155,67298
%N A298973 Squarefree primitive abundant numbers (first definition: having only deficient proper divisors).
%C A298973 Squarefree numbers (A005117) in A071395. The number of terms with n prime factors are counted in A295369. The subsequence of odd terms is A249263.
%C A298973 Two variants of the present sequence are possible: the terms listed by size, or as a table whose n-th row gives all those with n prime factors (so that A295369 would be the row lengths). They would differ only from a(322) = 692835 on, which is the first term with 6 prime factors, while a(755) = 4199030 is the last term with 5 prime factors.
%C A298973 A subsequence of the variant A249242, squarefree primitive abundant numbers using the 2nd definition, A091191, i.e., having no abundant proper divisors.
%C A298973 These numbers are also primitive unitary abundant numbers: unitary abundant numbers (A034683) that are also primitive abundant numbers (A071395). A unitary abundant number k is primitive if and only if usigma(k) - 2*k < 2*k/p^e, where p^e is the largest prime power dividing k and usigma is the sum of unitary divisors function (A034448). For numbers k in this sequence limsup_{k->oo} usigma(k)/k = 2. (Prasad and Reddy, 1990). - _Amiram Eldar_, Jul 18 2020
%D A298973 József Sándor, Dragoslav S. Mitrinovic and Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, chapter III, p. 115.
%H A298973 Amiram Eldar, <a href="/A298973/b298973.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from M. F. Hasler)
%H A298973 V. Siva Rama Prasad and D. Ram Reddy, <a href="http://insa.nic.in/writereaddata/UpLoadedFiles/IJPAM/20005a58_40.pdf">On primitive unitary abundant numbers</a>, Indian J. Pure Appl. Math., Vol. 21, No. 1 (1990), pp. 40-44.
%H A298973 D. P. Shukla and Shikha Yadav, <a href="https://eudml.org/doc/292063">Composition of arithmetical functions with generalization of perfect and related numbers</a>, Commentationes Mathematicae, Vol. 52, No. 2 (2012), pp. 153-170.
%e A298973 The only squarefree primitive abundant number (SFPAN) with only 3 prime factors is a(1) = 2*5*7 = 70. Indeed, this number is abundant (sigma(70) - 70 = 1 + 2 + 5 + 7 + 10 + 14 + 35 = 74) but all of 2*5, 2*7 and 5*7 are deficient. This is also the smallest (thus primitive) weird number, see A002975.
%e A298973 The A295369(4) = 18 SFPAN with 4 prime factors range from a(2) = 2*5*11*13 = 1430 to a(19) = 2*5*11*53 = 5830.
%e A298973 The A295369(5) = 610 SFPAN with 5 prime factors range from a(20) = 3*5*7*11*13 = 15015 to a(755) = 2*5*11*59*647 = 4199030, but the first term with 6 prime factors occurs already at a(322) = 3*5*11*13*17*19 = 692835.
%t A298973 spaQ[n_] := SquareFreeQ[n] && DivisorSigma[1, n] > 2*n && AllTrue[Most @ Divisors[n], DivisorSigma[1, #] < 2*# &]; Select[Range[70000], spaQ] (* _Amiram Eldar_, Jul 18 2020 *)
%o A298973 (PARI) is_A298973(n)=issquarefree(n)&&is_A071395(n)
%Y A298973 Cf. A005117 (squarefree numbers), A071395 (primitive abundant numbers, first definition), A091191 (idem, second definition), A249242 (squarefree numbers in A091191).
%Y A298973 Cf. A034448, A034683.
%K A298973 nonn
%O A298973 1,1
%A A298973 _M. F. Hasler_, Feb 16 2018