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A188342 Smallest odd primitive abundant number (A006038) having n distinct prime factors. 7

%I

%S 945,3465,15015,692835,22309287,1542773001,33426748355,1635754104985,

%T 114761064312895,9316511857401385,879315530560980695

%N Smallest odd primitive abundant number (A006038) having n distinct prime factors.

%C 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.

%C a(14) <= 88452776289145528645. - _Donovan Johnson_, Mar 31 2011

%C 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

%C 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

%H L. E. Dickson, <a href="http://www.jstor.org/stable/2370405">Finiteness of the odd perfect and primitive abundant numbers with n distinct prime factors</a>, American Journal of Mathematics 35 (1913), pp. 413-422.

%H H. N. Shapiro, <a href="http://projecteuclid.org/euclid.bams/1183513752">Note on a theorem of Dickson</a>, Bull Amer. Math. Soc. 55 (4) (1949), 450-452

%e From _M. F. Hasler_, Jul 17 2016: (Start)

%e 945 = 3^3 * 5 * 7

%e 3465 = 3^2 * 5 * 7 * 11

%e 15015 = 3 * 5 * 7 * 11 * 13

%e 692835 = 3 * 5 * 11 * 13 * 17 * 19 (n=6: gpf increases by 2 primes)

%e 22309287 = 3 * 7 * 11 * 13 * 17 * 19 * 23

%e 1542773001 = 3 * 7 * 11 * 17 * 19 * 23 * 29 * 31

%e 33426748355 = 5 * 7 * 11 * 13 * 17 * 19 * 23 * 29 * 31

%e 1635754104985 = 5 * 7 * 11 * 13 * 17 * 19 * 23 * 29 * 37 * 41 (here too)

%e 114761064312895 = 5 * 7 * 11 * 13 * 17 * 23 * 29 * 31 * 37 * 41 * 43

%e 9316511857401385 = 5 * 7 * 13 * 17 * 19 * 23 * 29 * 31 * 37 * 41 * 43 * 47

%e 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)

%t PrimAbunQ[n_] := Module[{x, y},

%t y = Most[Divisors[n]]; x = DivisorSigma[1, y];

%t DivisorSigma[1, n] > 2 n && AllTrue[x/y, # <= 2 &]];

%t Table[k = 1;

%t While[! PrimAbunQ[k] || Length[FactorInteger[k][[All, 1]]] != n,

%t k += 2]; k, {n, 3, 6}] (* _Robert Price_, Sep 26 2019 *)

%o (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

%K nonn,more

%O 3,1

%A _T. D. Noe_, Mar 28 2011

%E a(8)-a(12) from _Donovan Johnson_, Mar 29 2011

%E a(13) from _Donovan Johnson_, Mar 31 2011

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Last modified April 4 05:15 EDT 2020. Contains 333212 sequences. (Running on oeis4.)