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

%I A188342 #50 Feb 10 2024 09:05:43
%S A188342 945,3465,15015,692835,22309287,1542773001,33426748355,1635754104985,
%T A188342 114761064312895,9316511857401385,879315530560980695,
%U A188342 88452776289145528645,2792580508557308832935,428525983200229616718445,42163230434005200984080045,1357656019974967471687377449
%N A188342 Smallest odd primitive abundant number (A006038) having n distinct prime factors.
%C A188342 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 A188342 a(14) <= 88452776289145528645. - _Donovan Johnson_, Mar 31 2011
%C A188342 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 A188342 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
%C A188342 Comment from _Don Reble_, Jan 17 2023: (Start)
%C A188342 "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)."
%C A188342 This conjecture is correct up to a(50). (End)
%H A188342 Daniel Suteu, <a href="/A188342/b188342.txt">Table of n, a(n) for n = 3..27</a>
%H A188342 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 A188342 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 A188342 From _M. F. Hasler_, Jul 17 2016: (Start)
%e A188342                945 = 3^3 * 5 * 7
%e A188342               3465 = 3^2 * 5 * 7 * 11
%e A188342              15015 = 3 * 5 * 7 * 11 * 13
%e A188342             692835 = 3 * 5 * 11 * 13 * 17 * 19     (n=6: gpf increases by 2 primes)
%e A188342           22309287 = 3 * 7 * 11 * 13 * 17 * 19 * 23
%e A188342         1542773001 = 3 * 7 * 11 * 17 * 19 * 23 * 29 * 31
%e A188342        33426748355 = 5 * 7 * 11 * 13 * 17 * 19 * 23 * 29 * 31
%e A188342      1635754104985 = 5 * 7 * 11 * 13 * 17 * 19 * 23 * 29 * 37 * 41     (here too)
%e A188342    114761064312895 = 5 * 7 * 11 * 13 * 17 * 23 * 29 * 31 * 37 * 41 * 43
%e A188342   9316511857401385 = 5 * 7 * 13 * 17 * 19 * 23 * 29 * 31 * 37 * 41 * 43 * 47
%e A188342 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 A188342 PrimAbunQ[n_] := Module[{x, y},
%t A188342    y = Most[Divisors[n]]; x = DivisorSigma[1, y];
%t A188342    DivisorSigma[1, n] > 2 n  &&  AllTrue[x/y, # <= 2  &]];
%t A188342 Table[k = 1;
%t A188342  While[! PrimAbunQ[k] || Length[FactorInteger[k][[All, 1]]] != n,
%t A188342 k += 2]; k, {n, 3, 6}] (* _Robert Price_, Sep 26 2019 *)
%o A188342 (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
%o A188342 (PARI)
%o A188342 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)));
%o A188342 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
%Y A188342 Cf. A006038, A275449.
%K A188342 nonn
%O A188342 3,1
%A A188342 _T. D. Noe_, Mar 28 2011
%E A188342 a(8)-a(12) from _Donovan Johnson_, Mar 29 2011
%E A188342 a(13) from _Donovan Johnson_, Mar 31 2011
%E A188342 a(14)-a(17) confirmed and a(18) from _Daniel Suteu_, Feb 10 2024