A051015 Zeisel numbers.

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

Offset: 1

Author

Eric W. Weisstein

Keywords

• nonn

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_)
• Kevin S. Brown, Zeisel Numbers, MathPages website.
• OEIS Wiki, Zeisel numbers.
• Eric Weisstein's World of Mathematics, Zeisel Number.
• Wikipedia, Zeisel number.
• Helmut Zeisel, Primes of the form 2^(k-1)+k, sci.math newsgroup, February 24, 1994.

Crossrefs

Cf. A027746, A061422, A252094 (A values), A252095 (B values).

Programs

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

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 *)

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

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