OFFSET
3,1
COMMENTS
Dickson proves that there are only a finite number of odd primitive abundant numbers having n distinct prime factors. For n=3, there are 8 such numbers: 945, 1575, 2205, 7425, 78975, 131625, 342225, 570375. See A188439.
a(14) <= 88452776289145528645. - Donovan Johnson, Mar 31 2011
a(15) <= 2792580508557308832935, a(16) <= 428525983200229616718445, a(17) <= 42163230434005200984080045. If these a(n) are squarefree and don't have a greatest prime factor more than 3 primes away from that of the preceding term, then these bounds are the actual values of a(n). The PARI code needs only fractions of a second to compute further bounds, which under the given hypotheses are the actual values of a(n). - M. F. Hasler, Jul 17 2016
It appears that the terms are squarefree for n >= 5, so they yield also the smallest term of A249263 with n factors; see A287581 for the largest such, and A287590 for the number of such terms with n factors. (For nonsquarefree odd abundant numbers, this seems to be known only for n = 3 and n = 4 prime factors (8 respectively 576 terms), cf. A188439.) - M. F. Hasler, May 29 2017
Comment from Don Reble, Jan 17 2023: (Start)
"If these a(n) are squarefree and don't have a greatest prime factor more than 3 primes away from that of the preceding term, then these bounds are the actual values of a(n)."
This conjecture is correct up to a(50). (End)
LINKS
Daniel Suteu, Table of n, a(n) for n = 3..27
L. E. Dickson, Finiteness of the odd perfect and primitive abundant numbers with n distinct prime factors, American Journal of Mathematics 35 (1913), pp. 413-422.
H. N. Shapiro, Note on a theorem of Dickson, Bull Amer. Math. Soc. 55 (4) (1949), 450-452
EXAMPLE
From M. F. Hasler, Jul 17 2016: (Start)
945 = 3^3 * 5 * 7
3465 = 3^2 * 5 * 7 * 11
15015 = 3 * 5 * 7 * 11 * 13
692835 = 3 * 5 * 11 * 13 * 17 * 19 (n=6: gpf increases by 2 primes)
22309287 = 3 * 7 * 11 * 13 * 17 * 19 * 23
1542773001 = 3 * 7 * 11 * 17 * 19 * 23 * 29 * 31
33426748355 = 5 * 7 * 11 * 13 * 17 * 19 * 23 * 29 * 31
1635754104985 = 5 * 7 * 11 * 13 * 17 * 19 * 23 * 29 * 37 * 41 (here too)
114761064312895 = 5 * 7 * 11 * 13 * 17 * 23 * 29 * 31 * 37 * 41 * 43
9316511857401385 = 5 * 7 * 13 * 17 * 19 * 23 * 29 * 31 * 37 * 41 * 43 * 47
879315530560980695 = 5 * 7 * 13 * 17 * 19 * 23 * 29 * 31 * 37 * 41 * 53 * 59 * 61 (n=13: gpf increases for the first time by 3 primes) (End)
MATHEMATICA
PrimAbunQ[n_] := Module[{x, y},
y = Most[Divisors[n]]; x = DivisorSigma[1, y];
DivisorSigma[1, n] > 2 n && AllTrue[x/y, # <= 2 &]];
Table[k = 1;
While[! PrimAbunQ[k] || Length[FactorInteger[k][[All, 1]]] != n,
k += 2]; k, {n, 3, 6}] (* Robert Price, Sep 26 2019 *)
PROG
(PARI) A188342=[0, 0, 945, 3465]; a(n, D(n)=n\6+1)={while(n>#A188342, my(S=#A188342, T=factor(A188342[S])[, 1], M=[primepi(T[1]), primepi(T[#T])+D(S++)], best=prime(M[2])^S); forvec(v=vector(S, i, M), best>(T=prod(i=1, #v, prime(v[i]))) && (S=prod(i=1, #v, prime(v[i])+1)-T*2)>0 && S*prime(v[#v])<T*2 && best=T, 2); A188342=concat(A188342, best)); A188342[n]} \\ Assuming a(n) squarefree for n>4, search is exhaustive within the limit primepi(gpf(a(n))) <= primepi(gpf(a(n-1)))+D(n), with D(n) given as optional 2nd arg. - M. F. Hasler, Jul 17 2016
(PARI)
generate(A, B, n) = A=max(A, vecprod(primes(n+1))\2); (f(m, p, j) = my(list=List()); if(sigma(m) > 2*m, return(list)); forprime(q=p, sqrtnint(B\m, j), my(v=m*q); while(v <= B, if(j==1, if(v>=A && sigma(v) > 2*v, my(F=factor(v)[, 1], ok=1); for(i=1, #F, if(sigma(v\F[i], -1) > 2, ok=0; break)); if(ok, listput(list, v))), if(v*(q+1) <= B, list=concat(list, f(v, q+1, j-1)))); v *= q)); list); vecsort(Vec(f(1, 3, n)));
a(n) = my(x=vecprod(primes(n+1))\2, 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
T. D. Noe, Mar 28 2011
EXTENSIONS
a(8)-a(12) from Donovan Johnson, Mar 29 2011
a(13) from Donovan Johnson, Mar 31 2011
a(14)-a(17) confirmed and a(18) from Daniel Suteu, Feb 10 2024
STATUS
approved