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A051015
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Zeisel numbers.
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9
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105, 1419, 1729, 1885, 4505, 5719, 15387, 24211, 25085, 27559, 31929, 54205, 59081, 114985, 207177, 208681, 233569, 287979, 294409, 336611, 353977, 448585, 507579, 982513, 1012121, 1073305, 1242709, 1485609, 2089257, 2263811, 2953711, 3077705, 3506371, 3655861, 3812599
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OFFSET
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1,1
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COMMENTS
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Pick any integers A and B and consider the linear recurrence relation given by p(0) = 1, p(i + 1) = A*p(i) + B. If for some n > 2, p(1), p(2), ..., p(n) are distinct primes, then the product of these primes is called a Zeisel number.
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LINKS
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MATHEMATICA
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maxTerm = 3*10^7; ZeiselQ[n_] := Module[{a, b, pp, eq, r}, If[PrimeQ[n] || ! SquareFreeQ[n], False, pp = Join[{1}, FactorInteger[n][[All, 1]]]; If[Length[pp] <= 3, False, eq = Thread[Rest[pp] == b + a*Most[pp]]; r = Reduce[eq, {a, b}, Integers]; r =!= False]]]; p = 3; A051015 = Reap[While[p^3 < maxTerm, q = NextPrime[p]; While[p*q^2 < maxTerm, If[ ! IntegerQ[a = (q - p)/(p - 1)] || !IntegerQ[b = (p^2 - q)/(p - 1)], q = NextPrime[q]; Continue[]]; r = b + a*q; n = r*p*q; While[PrimeQ[r] && n < maxTerm, Sow[n]; r = b + a*r; n *= r]; q = NextPrime[q]]; p = NextPrime[p]]][[2, 1]]; A051015 = Select[Sort[A051015], ZeiselQ] (* Jean-François Alcover, Oct 31 2012, with much help from Giovanni Resta *)
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PROG
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(PARI) is_A051015(n)={#(n=factor(n)~)>2 & vecmax(n[2, ])==1 & denominator(n[2, 1]=(n[1, 3]-n[1, 2])/(n[1, 2]-n[1, 1]))==1 & #Set(n[1, ]-n[2, 1]*concat(1, vecextract(n[1, ], "^-1")))==1} \\ - M. F. Hasler, Oct 31 2012
(Haskell)
a051015 n = a051015_list !! (n-1)
a051015_list = filter zeisel [3, 5 ..] where
zeisel x = 0 `notElem` ds && length ds > 2 &&
all (== 0) (zipWith mod (tail ds) ds) && all (== q) qs
where q:qs = (zipWith div (tail ds) ds)
ds = zipWith (-) (tail ps) ps
ps = 1 : a027746_row x
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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