A287581 Largest squarefree odd primitive abundant number with n prime factors.

442365, 13455037365, 1725553747427327895, 977844705701880720314685634538055, 29094181301361888360228876470808927597684302024968488289496445

Offset: 5

Author

M. F. Hasler, May 26 2017

Keywords

• nonn

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.

Examples

			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.
		

Crossrefs

Cf. A287590, A249263.

Programs

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}

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.