OFFSET
5,1
COMMENTS
See A188342 = (945, 3465, 15015, 692835, 22309287, ...) for the least odd primitive abundant number (A006038) with n distinct prime factors.
At least up to a(11), the greatest prime factor gpf(a(n)) = Q(a(n)/gpf(a(n))), where Q(N) = floor(sigma(N)/(2N-sigma(N))). In general one has to apply the precprime() function A007917 to this integer.
The above holds also for a(12)-a(15). Lars Blomberg, Apr 09 2018
EXAMPLE
We have: a(5) = 945 = 3^3 * 5 * 7,
a(6) = 7425 = 3^3 * 5^2 * 11,
a(7) = 81081 = 3^4 * 7 * 11 * 13,
a(8) = 78975 = 3^5 * 5^2 * 13,
a(9) = 1468935 = 3^6 * 5 * 13 * 31,
a(10) = 6375105 = 3^7 * 5 * 11 * 53,
a(11) = 85930875 = 3^6 * 5^3 * 23 * 41,
a(12) = 307879299 = 3^7 * 7^2 * 13^2 * 17,
a(13) = 1519691625 = 3^8 * 5^3 * 17 * 109,
a(14) = 8853249375 = 3^8 * 5^4 * 17 * 127,
a(15) = 17062700625 = 3^9 * 5^4 * 19 * 73.
PROG
(PARI) a(n)=for(i=1, #A=A006038, bigomega(A[i])==n&&return(A[i])) \\ Provided that A006038 is defined as a set with enough elements. - M. F. Hasler, Jul 27 2016
(PARI)
generate(A, B, n) = A=max(A, 3^n); (f(m, p, k) = my(list=List()); if(sigma(m) > 2*m, return(list)); if(k==1, forprime(q=max(p, ceil(A/m)), B\m, my(t=m*q); if(sigma(t) > 2*t, my(F=factor(t)[, 1], ok=1); for(i=1, #F, if(sigma(t\F[i], -1) > 2, ok=0; break)); if(ok, listput(list, t)))), forprime(q = p, sqrtnint(B\m, k), list=concat(list, f(m*q, q, k-1)))); list); vecsort(Vec(f(1, 3, n)));
a(n) = my(x=3^n, y=2*x); while(1, my(v=generate(x, y, n)); if(#v >= 1, return(v[1])); x=y+1; y=2*x); \\ Daniel Suteu, Feb 10 2024
CROSSREFS
KEYWORD
nonn
AUTHOR
M. F. Hasler, Jul 27 2016
EXTENSIONS
a(12)-a(15) from Lars Blomberg, Apr 09 2018
a(16)-a(25) from Daniel Suteu, Feb 10 2024
STATUS
approved