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A182864 a(n) is the smallest number such that a(n)^2 +1 = (a(n-1)^2 + 1)*q(n) where a(0)=0 and q(n) is a sequence of increasing prime numbers. 0
0, 1, 3, 13, 123, 2903, 3373867, 895293820337 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

Table of n, a(n) for n=0..7.

EXAMPLE

a(0) = 0;

a(1) = 1 because a(1)^2+1 = (0^2 + 1)*2 = 2;

a(2) = 3 because a(2)^2 + 1 = (1^2 + 1)*5 = 2*5;

a(3) = 13 because a(3)^2 + 1 = (3^2+1)*17 =2*5*17;

a(4) = 123 because 123^2 + 1 = (13^2 + 1)*89 = 2*5*17*89;

a(5) = 2903 because 2903^2 + 1 = (123^2+1)*557 = 2*5*17*89*557;

a(6) = 3373867 because 3373867^2+1 = 2 * 5 * 17 * 89 * 557 * 1350709.

MAPLE

with(numtheory):nn:=100000:T:=array(1..nn):k:=1: for x from 1 to nn do: p:=4*x+1:if

  type(p, prime)=true then T[k]:=p:k:=k+1:else fi:od:pp:=2:for u from 1 to k do:id:=0:for

  x from 1 to nn while(id=0) do:y:=x^2+1:z:=pp*T[u]:if y=z then id:=1:print(x):else

  fi:od:if id=1 then pp:=y:else fi:od:

MATHEMATICA

a[0] = 0; a[1] = 1; lst = {2}; a[n_] := a[n] = (s = 1 + a[n - 1]^2; p = NextPrime[lst[[-1]]]; While[q = Sqrt[p*s - 1]; ! IntegerQ@ q, p = NextPrime@ p]; AppendTo[lst, p]; q) (* Robert G. Wilson v, Feb 06 2011 *)

CROSSREFS

Sequence in context: A191955 A241458 A302861 * A208590 A228648 A161677

Adjacent sequences:  A182861 A182862 A182863 * A182865 A182866 A182867

KEYWORD

nonn,hard

AUTHOR

Michel Lagneau, Feb 01 2011

EXTENSIONS

a(7) from Donovan Johnson, Feb 11 2011

STATUS

approved

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Last modified June 12 10:58 EDT 2021. Contains 344947 sequences. (Running on oeis4.)