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%I #20 Nov 21 2021 08:15:05
%S 3,39,225,249,321,447,471,519,681,831,921,993,1119,1191,1473,1641,
%T 1671,1857,1929,1983,2361,2391,2463,2625,2631,2913,3321,3369,3561,
%U 3591,3777,3807,3831,3903,4119,4281,4287,4359,4545,4569,4791,5001,5025,5079,5241,5481
%N Numbers m such that both m^2-1 and m^2 are refactorable numbers (A033950).
%C Numbers m such that m^2-1 is divisible by d(m^2-1) and m^2 is divisible by d(m^2), d = A000005.
%C Zelinsky (2002, Theorem 59, p. 15) proved that if k > 1, k and k+1 are both refactorable numbers, then k is even. Such k must be of the form m^2-1 for some odd m.
%C The smallest term not divisible by 3 is a(66) = 9025.
%C For the first terms we have d(a(n)^2-1) > d(a(n)^2). But this is not always the case. The smallest counterexample is a(30) = 3591, where d(3591^2-1) = 40 and d(3591^2) = 63. The terms m such that d(m^2-1) < d(m^2) are listed in A342970. [Note that d(m^2-1) = d(m^2) is impossible since d(m^2-1) is even and d(m^2) is odd. - _Jianing Song_, Nov 21 2021]
%H Jianing Song, <a href="/A342969/b342969.txt">Table of n, a(n) for n = 1..3110</a> (all terms <= 10^6).
%H Joshua Zelinsky, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL5/Zelinsky/zelinsky9.html">Tau Numbers: A Partial Proof of a Conjecture and Other Results</a>, Journal of Integer Sequences, Vol. 5 (2002), Article 02.2.8.
%F A036898(2*n+1) = A114617(n+1) = a(n)^2 - 1; A036898(2*n+2) = A114617(n+1) + 1 = a(n)^2.
%e 39 is a term since 39^2-1 = 1520 is divisible by d(1520) = 20 and 39^2 = 1521 is divisible by d(1521) = 9.
%o (PARI) isrefac(n) = ! (n % numdiv(n));
%o isA342969(n) = (n>1) && isrefac(n^2-1) && isrefac(n^2)
%Y Cf. A000005, A033950, A036896, A036898, A114617, A342970.
%K nonn,easy
%O 1,1
%A _Jianing Song_, Apr 01 2021
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