cp's OEIS Frontend

A342969 Numbers m such that both m^2-1 and m^2 are refactorable numbers (A033950).

%I A342969 #26 Feb 03 2025 02:11:47
%S A342969 3,39,225,249,321,447,471,519,681,831,921,993,1119,1191,1473,1641,
%T A342969 1671,1857,1929,1983,2361,2391,2463,2625,2631,2913,3321,3369,3561,
%U A342969 3591,3777,3807,3831,3903,4119,4281,4287,4359,4545,4569,4791,5001,5025,5079,5241,5481
%N A342969 Numbers m such that both m^2-1 and m^2 are refactorable numbers (A033950).
%C A342969 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 A342969 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 A342969 The smallest term not divisible by 3 is a(66) = 9025.
%C A342969 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 A342969 Amiram Eldar, <a href="/A342969/b342969.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..3110 from Jianing Song)
%H A342969 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 A342969 A036898(2*n+1) = A114617(n+1) = a(n)^2 - 1; A036898(2*n+2) = A114617(n+1) + 1 = a(n)^2.
%e A342969 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.
%t A342969 refQ[n_] := Divisible[n, DivisorSigma[0, n]]; Select[Range[6000], And @@ refQ /@ (#^2 - {1, 0}) &] (* _Amiram Eldar_, Feb 03 2025 *)
%o A342969 (PARI) isrefac(n) = ! (n % numdiv(n));
%o A342969 isA342969(n) = (n>1) && isrefac(n^2-1) && isrefac(n^2)
%Y A342969 Cf. A000005, A033950, A036896, A036898, A114617, A342970.
%K A342969 nonn,easy
%O A342969 1,1
%A A342969 _Jianing Song_, Apr 01 2021