A298973 Squarefree primitive abundant numbers (first definition: having only deficient proper divisors).

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

Offset: 1

Author

M. F. Hasler, Feb 16 2018

Keywords

• nonn

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 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.

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

Examples

			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.
		

References

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

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from M. F. Hasler)
• V. Siva Rama Prasad and D. Ram Reddy, On primitive unitary abundant numbers, Indian J. Pure Appl. Math., Vol. 21, No. 1 (1990), pp. 40-44.
• D. P. Shukla and Shikha Yadav, Composition of arithmetical functions with generalization of perfect and related numbers, Commentationes Mathematicae, Vol. 52, No. 2 (2012), pp. 153-170.

Crossrefs

Cf. A005117 (squarefree numbers), A071395 (primitive abundant numbers, first definition), A091191 (idem, second definition), A249242 (squarefree numbers in A091191).

Cf. A034448, A034683.

Programs

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 *)

PARI

is_A298973(n)=issquarefree(n)&&is_A071395(n)