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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

Eric W. Weisstein

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

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Last modified January 20 19:53 EST 2020. Contains 331096 sequences. (Running on oeis4.)