|
%I #13 Feb 10 2024 12:05:52
%S 945,7425,81081,78975,1468935,6375105,85930875,307879299,1519691625,
%T 8853249375,17062700625,535868474337,2241870572475,12759034818375,
%U 64260996890625,866566808687853,2964430488515625,23849823423763953,100139192108634825,772934641006640625,2696807941801171875
%N Least odd primitive abundant number with n prime factors, counted with multiplicity.
%C See A188342 = (945, 3465, 15015, 692835, 22309287, ...) for the least odd primitive abundant number (A006038) with n distinct prime factors.
%C 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.
%C The above holds also for a(12)-a(15). _Lars Blomberg_, Apr 09 2018
%e We have: a(5) = 945 = 3^3 * 5 * 7,
%e a(6) = 7425 = 3^3 * 5^2 * 11,
%e a(7) = 81081 = 3^4 * 7 * 11 * 13,
%e a(8) = 78975 = 3^5 * 5^2 * 13,
%e a(9) = 1468935 = 3^6 * 5 * 13 * 31,
%e a(10) = 6375105 = 3^7 * 5 * 11 * 53,
%e a(11) = 85930875 = 3^6 * 5^3 * 23 * 41,
%e a(12) = 307879299 = 3^7 * 7^2 * 13^2 * 17,
%e a(13) = 1519691625 = 3^8 * 5^3 * 17 * 109,
%e a(14) = 8853249375 = 3^8 * 5^4 * 17 * 127,
%e a(15) = 17062700625 = 3^9 * 5^4 * 19 * 73.
%o (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
%o (PARI)
%o 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)));
%o 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
%Y Cf. A006038, A188342, A188439.
%K nonn
%O 5,1
%A _M. F. Hasler_, Jul 27 2016
%E a(12)-a(15) from _Lars Blomberg_, Apr 09 2018
%E a(16)-a(25) from _Daniel Suteu_, Feb 10 2024
|
