

A051015


Zeisel numbers.


5



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

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)
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] (* JeanFrançois 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


CROSSREFS

Sequence in context: A033593 A058844 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 JeanFrançois Alcover and doublechecked by M. F. Hasler, Oct 31 2012
Values < 10^15 by Lars Blomberg, Nov 02 2012


STATUS

approved



