cp's OEIS Frontend

A347193 a(n) is the smallest m such that A347191(m) = 2*n, where A347191(m) = tau(m^2 - 1).

%I A347193 #34 May 15 2022 23:50:11
%S A347193 2,3,8,5,7,15,33,11,17,23,513,19,13841287200,31,73,29,
%T A347193 650377879817809571042122834560,49,131073,41,97,1537,31381059608,79,
%U A347193 50626,10239,127,223,459986536544739960976800,71,8193465725814765556554001028792218848,109,61953,163839,161
%N A347193 a(n) is the smallest m such that A347191(m) = 2*n, where A347191(m) = tau(m^2 - 1).
%C A347193 Generalization of the questions proposed in Diophante problem A1885 (see links).
%C A347193 When p is prime, as a(p) is the smallest m such that tau(m^2 - 1) = 2*p, hence m^2 - 1 is of the form q * r^(p-1) with q > r primes, so we must solve the Diophantine equation (m-1)*(m+1) = q * r^(p-1) to get the smallest m, when only p is known.
%C A347193 Two cases must be checked:
%C A347193 -> r = 2: if 2^(p-3) + 1 is prime, then m = 2^(p-2) + 1, and
%C A347193           if 2^(p-3) - 1 is prime, then m = 2^(p-2) - 1.
%C A347193 If there is a solution m in the case r = 2, then a(p) is this smallest solution m (see example a(11)); if there is no solution m with r = 2, then try the 2nd case.
%C A347193 -> r odd prime: if r^(p-1) + 2 is prime, then m = r^(p-1) + 1, and
%C A347193                 if r^(p-1) - 2 is prime, then m = r^(p-1) - 1 (example a(13)).
%H A347193 Diophante, <a href="http://www.diophante.fr/problemes-par-themes/arithmetique-et-algebre/a1-pot-pourri/3848-a1885-caches-derriere-leurs-diviseurs">A1885, Cachés derrière leurs diviseurs</a> (in French).
%H A347193 Adrian Dudek, <a href="http://arxiv.org/abs/1507.08893">On the Number of Divisors of n^2-1</a>, arXiv:1507.08893 [math.NT], 2015.
%e A347193 tau(2^2 - 1) = 2 = 2*1, so a(1) = 2.
%e A347193 tau(3^2 - 1) = 4 = 2*2, so a(2) = 3.
%e A347193 tau(4^2 - 1) = 4 = 2*2, tau(5^2 - 1) = 8 = 2*4 so a(4) = 5.
%e A347193 For a(11): if r = 2, 2^8 + 1 = 257 is prime, while 2^8 - 1 is not prime, hence a(11) = 2^9 + 1 = 513.
%e A347193 For a(13):
%e A347193   if r = 2, 2^10 +- 1 are not prime, so not possible;
%e A347193   if r = 3, 3^12 +- 2 are not prime, so not possible;
%e A347193   if r = 5, 5^12 +- 2 are not prime, so not possible;
%e A347193   if r = 7, 7^12 - 2 = 13841287199 is prime, while 7^12 + 2 is not prime, then a(13) = 13841287199+1 = 13841287200.
%Y A347193 Cf. A000005, A347191, A347192, A347194.
%K A347193 nonn
%O A347193 1,1
%A A347193 _Bernard Schott_, Sep 17 2021
%E A347193 a(31)-a(35) from _Jinyuan Wang_, Sep 23 2021