cp's OEIS Frontend

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

%I A287581 #13 Jun 15 2017 01:19:39
%S A287581 442365,13455037365,1725553747427327895,
%T A287581 977844705701880720314685634538055,
%U A287581 29094181301361888360228876470808927597684302024968488289496445
%N A287581 Largest squarefree odd primitive abundant number with n prime factors.
%C A287581 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.
%C A287581 See A287590 for the number of squarefree odd primitive abundant numbers (A249263) with n prime factors.
%C A287581 The next term, a(10), is too large to display.
%C A287581 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.
%F A287581 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.
%e A287581 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.
%e A287581 a(6) = 13455037365 = 3 * 5 * 7 * 11 * 389 * 29947,
%e A287581 a(7) = 1725553747427327895 = 3 * 5 * 7 * 11 * 389 * 29959 * 128194559,
%e A287581 a(8) = 3 * 5 * 7 * 11 * 389 * 29959 * 128194589 * 566684450325179,
%e A287581 a(9) = a(8)/gpf(a(8)) * 566684450325197 * 29753376105337343078941364893,
%e A287581 a(10) = a(9)/gpf(a(9)) * 29753376105337343078941364947 * 30082232218581187462432471034748868284388270918928732059.
%o A287581 (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}
%Y A287581 Cf. A287590, A249263.
%K A287581 nonn
%O A287581 5,1
%A A287581 _M. F. Hasler_, May 26 2017