

A298973


Squarefree primitive abundant numbers (first definition: having only deficient proper divisors).


3



70, 1430, 1870, 2002, 2090, 2210, 2470, 2530, 2990, 3190, 3230, 3410, 3770, 4030, 4070, 4510, 4730, 5170, 5830, 15015, 19635, 21945, 23205, 25935, 26565, 31395, 33495, 33915, 35805, 39585, 41055, 42315, 42735, 45885, 47355, 49665, 49742, 50505, 51765, 54285, 55965, 58695, 58786, 60214, 61215, 64155, 67298
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OFFSET

1,1


COMMENTS

Squarefree numbers (A005117) in A071395. The number of terms with n prime factors are counted in A295369. The subsequence of odd terms is A249263.
Two variants of the present sequence are possible: the terms listed by size, or as a table whose nth 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.
A subsequence of the variant A249242, squarefree primitive abundant numbers using the 2nd definition, A091191, i.e., having no abundant proper divisors.
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


REFERENCES

József Sándor, Dragoslav S. Mitrinovic and Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, chapter III, p. 115.


LINKS



EXAMPLE

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.
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.
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.


MATHEMATICA

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 *)


PROG



CROSSREFS



KEYWORD

nonn


AUTHOR



STATUS

approved



