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 A188342 Smallest odd primitive abundant number (A006038) having n distinct prime factors. 7
 945, 3465, 15015, 692835, 22309287, 1542773001, 33426748355, 1635754104985, 114761064312895, 9316511857401385, 879315530560980695 (list; graph; refs; listen; history; text; internal format)
 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 LINKS 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), primepi(T[#T])+D(S++)], best=prime(M)^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])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 CROSSREFS Sequence in context: A127667 A252184 A188263 * A109729 A294025 A275449 Adjacent sequences:  A188339 A188340 A188341 * A188343 A188344 A188345 KEYWORD nonn,more 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 STATUS approved

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Last modified February 27 10:15 EST 2020. Contains 332304 sequences. (Running on oeis4.)