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A188342
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Smallest odd primitive abundant number (A006038) having n distinct prime factors.
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7
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945, 3465, 15015, 692835, 22309287, 1542773001, 33426748355, 1635754104985, 114761064312895, 9316511857401385, 879315530560980695
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listen;
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OFFSET
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3,1
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COMMENTS
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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
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LINKS
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Table of n, a(n) for n=3..13.
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
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EXAMPLE
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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)
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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Sequence in context: A127667 A252184 A188263 * A109729 A294025 A275449
Adjacent sequences: A188339 A188340 A188341 * A188343 A188344 A188345
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KEYWORD
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nonn,more
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AUTHOR
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T. D. Noe, Mar 28 2011
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EXTENSIONS
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a(8)-a(12) from Donovan Johnson, Mar 29 2011
a(13) from Donovan Johnson, Mar 31 2011
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STATUS
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approved
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