login
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
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)
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.
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, A061422, A252094 (A values), A252095 (B values).
Sequence in context: A058844 A297800 A165382 * A221449 A076377 A165374
KEYWORD
nonn
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