

A287581


Largest squarefree odd primitive abundant number with n prime factors.


5




OFFSET

5,1


COMMENTS

There is no squarefree odd abundant number with fewer than 5 prime factors: the largest abundancy an odd squarefree number with 4 prime factors can have is that of N = 3*5*7*11 with sigma_{1}(N) = sigma(N)/N = 2  2/385.
See A287590 for the number of squarefree odd primitive abundant numbers (A249263) with n prime factors.
The next term, a(10), is too large to display.
It appears that the largest odd primitive abundant number with a given number of prime factors counted with multiplicity (bigomega = A001222), is always squarefree. Whenever this holds for a given n, then a(n) is also equal to the last term in row n of A287646 which lists odd primitive abundant numbers with n prime factors.


LINKS

Table of n, a(n) for n=5..9.


FORMULA

a(n+1) = (a(n)/p(n))*p'(n)*q(n), where p(n) = gpf(a(n)), p'(n) = nextprime(p(n)+1), q(n) = precprime(1/(2/sigma[1](a(n)/p(n)*p'(n))1)), sigma[1](x) = sigma(x)/x; conjectured to hold for all n >= 5.


EXAMPLE

a(5) = 442365 = 3 * 5 * 7 * 11 * 383 is the largest squarefree odd primitive abundant number (SOPAN). Here, 3*5*7*11 is the smallest possibility to produce a squarefree odd deficient number with 4 prime factors, and it is the one with the largest possible abundancy, and 383 is the largest prime by which this can be multiplied to yield an abundant number. One can increase 11 up to 19 to get more SOPAN (for a total of 71 + 12 + 3 + 1 = 87 = A287590(5) SOPAN with 5 factors), none of which is larger. One can see that increasing the 3rd prime factor 7 to 11 yields no further possibilities, and therefore also the second and third factor can't be increased.
a(6) = 13455037365 = 3 * 5 * 7 * 11 * 389 * 29947,
a(7) = 1725553747427327895 = 3 * 5 * 7 * 11 * 389 * 29959 * 128194559,
a(8) = 3 * 5 * 7 * 11 * 389 * 29959 * 128194589 * 566684450325179,
a(9) = a(8)/gpf(a(8)) * 566684450325197 * 29753376105337343078941364893,
a(10) = a(9)/gpf(a(9)) * 29753376105337343078941364947 * 30082232218581187462432471034748868284388270918928732059.


PROG

(PARI) A287581(n, p=3, P=p, s=2)={forstep(i=n, 2, 1, n=max(1\(1+s/=1+1/p), p+1); P*=p=if(i>2, nextprime(n), precprime(n))); P}


CROSSREFS

Cf. A287590, A249263.
Sequence in context: A238069 A234150 A294987 * A230084 A250839 A270610
Adjacent sequences: A287578 A287579 A287580 * A287582 A287583 A287584


KEYWORD

nonn


AUTHOR

M. F. Hasler, May 26 2017


STATUS

approved



