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A261803 a(n) is the smallest number satisfying a(n)^2+1 = p1(n)*p2(n), p1(n) < p2(n) primes, such that p2(n+1)/p1(n+1) < p2(n)/p1(n) with the initial condition p2(1)/p1(1) < 3/2. 1

%I

%S 50,334,516,670,844,1164,1250,1800,2450,9800,14450,20000,24200,101250,

%T 105800,135200,162450,168200,204800,304200,336200,451250,480200,

%U 490050,530450,696200,924800,966050,1008200,1125000,1155200,1428050,1805000,2332800,2420000

%N a(n) is the smallest number satisfying a(n)^2+1 = p1(n)*p2(n), p1(n) < p2(n) primes, such that p2(n+1)/p1(n+1) < p2(n)/p1(n) with the initial condition p2(1)/p1(1) < 3/2.

%C The sequence is probably infinite.

%C A majority of numbers in the sequence are of the form 2*q^2 with q = 5, 25, 30, 35, 70, 85, 100, 110, 225, 230, 260, 285, 290, 320, 390, 410, ... So, it seems that {a(n)} = {334, 516, 670, 844, 1164} union {2*A109306(n)^2} where A109306 are the numbers k such that k^2 + (k-1)^2 and k^2 + (k+1)^2 are both primes.

%H Jean-Fran├žois Alcover, <a href="/A261803/b261803.txt">Table of n, a(n) for n = 1..500</a>

%e a(1) = 50 because 50^2+1 = 41*61 => 61/41 = 1.4878... < 1.5

%e a(2) = 334 because 334^2+1 = 281*397 => 397/281 = 1.4128... < 1.4878...

%e a(3) = 516 because 516^2+1 = 449*593 => 593/449 = 1.3207... < 1.4128...

%e a(4) = 670 because 670^2+1 = 593*757 => 757/593 = 1.2765... < 1.3207...

%p with(numtheory):nn:=100:d:=1.5:

%p for n from 1 to nn do:

%p x:=factorset(n^2+1):n0:=bigomega(n^2+1):

%p if n0=2

%p then

%p q:=evalf(x[2]/x[1]):

%p if q<d then

%p d:=q:printf(`%d, `,n):

%p else fi:

%p fi :

%p od:

%t (* Assumption: n>7 ==> a(n)=0 mod 50 *) a[n_] := a[n] = For[k = Which[n==1, 0, n <= 7, a[n-1]+1, True, a[n-1] + 50], True, k = Which[n <= 7, k+1, k == a[n-1]+1, k+49, True, k+50], f = FactorInteger[k^2+1]; If[Length[f] == 2, If[f[[All, 2]] == {1, 1}, {p1, q1} = f[[All, 1]]; If[q1/p1 < If[n == 1, 3/2, q[n-1]/p[n-1]], p[n] = p1; q[n] = q1; Return[k]]]]]; Table[Print["a(", n, ") = ", a[n], " p = ", p[n], " q = ", q[n], " q/p = ", N[q[n]/p[n], 10], " q-p = ", q[n]-p[n]]; a[n], {n, 1, 100}] (* _Jean-Fran├žois Alcover_, Sep 28 2015 *)

%Y Cf. A002496, A109306, A206328.

%Y Subsequence of A085722.

%K nonn

%O 1,1

%A _Michel Lagneau_, Sep 01 2015

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Last modified July 15 23:32 EDT 2020. Contains 335774 sequences. (Running on oeis4.)