A275449 Least odd primitive abundant number with n prime factors, counted with multiplicity.

945, 7425, 81081, 78975, 1468935, 6375105, 85930875, 307879299, 1519691625, 8853249375, 17062700625, 535868474337, 2241870572475, 12759034818375, 64260996890625, 866566808687853, 2964430488515625, 23849823423763953, 100139192108634825, 772934641006640625, 2696807941801171875

Offset: 5

Author

M. F. Hasler, Jul 27 2016

Keywords

• nonn

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

Examples

			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.
		

Crossrefs

Cf. A006038, A188342, A188439.

Programs

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

Extensions

a(12)-a(15) from Lars Blomberg, Apr 09 2018

a(16)-a(25) from Daniel Suteu, Feb 10 2024