This site is supported by donations to The OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A051015 Zeisel numbers. 9
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS 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. LINKS M. F. Hasler and Lars Blomberg, Table of n, a(n) for n = 1..9607 (first 70 terms from M. F. Hasler) OEIS Wiki, Zeisel numbers Eric Weisstein's World of Mathematics, Zeisel Number. Wikipedia, Zeisel number MATHEMATICA 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 *) PROG (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 -- Reinhard Zumkeller, Dec 15 2014 CROSSREFS Cf. A027746, A252094 (A values), A252095 (B values). Sequence in context: A058844 A297800 A165382 * A221449 A076377 A165374 Adjacent sequences:  A051012 A051013 A051014 * A051016 A051017 A051018 KEYWORD nonn AUTHOR EXTENSIONS More terms from David Wasserman, Feb 19 2002 Extended by Karsten Meyer, Jun 08 2006, but values were incorrect. M. F. Hasler, Oct 31 2012 Values up to a(70) computed by Jean-François Alcover and double-checked by M. F. Hasler, Oct 31 2012 Values < 10^15 by Lars Blomberg, Nov 02 2012 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified October 15 18:18 EDT 2019. Contains 328037 sequences. (Running on oeis4.)