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A342970 Numbers m such that both m^2-1 and m^2 are refactorable numbers (A033950) and that m^2 has more divisors than m^2-1.

%I A342970 #15 Feb 03 2025 02:11:08
%S A342970 3591,4545,5481,6975,8415,9639,11319,11583,11745,12225,12735,16065,
%T A342970 18711,24255,24759,30015,31671,39105,40257,41535,41769,44631,44865,
%U A342970 52065,52569,53055,54975,56511,60255,60705,64071,64575,69825,72009,73665,76095,81081,81855,87129
%N A342970 Numbers m such that both m^2-1 and m^2 are refactorable numbers (A033950) and that m^2 has more divisors than m^2-1.
%C A342970 Numbers m such that m^2-1 is divisible by d(m^2-1), m^2 is divisible by d(m^2) and d(m^2) > d(m^2-1), d = A000005.
%C A342970 The smallest term not divisible by 3 is a(1048) = 2907025.
%H A342970 Amiram Eldar, <a href="/A342970/b342970.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1500 from Jianing Song)
%e A342970 5481 is a term since 5481^2-1 is divisible by d(5481^2-1) = 40, 5481^2 is divisible by d(5481^2) = 63, and 63 > 40.
%e A342970 2907025 is a term since 2907025^2-1 is divisible by d(2907025^2-1) = 96, 2907025^2 is divisible by d(2907025^2) = 125, and 125 > 96.
%t A342970 q[n_] := Module[{d = DivisorSigma[0, n^2 - {1, 0}]}, d[[1]] < d[[2]] && Divisible[n^2-1, d[[1]]] && Divisible[n^2, d[[2]]]]; Select[Range[10^5], q] (* _Amiram Eldar_, Feb 03 2025 *)
%o A342970 (PARI) isA342970(n) = if(n>1, my(d1 = numdiv(n^2-1), d2 = numdiv(n^2)); !((n^2-1) % d1) && !(n^2 % d2) && d2 > d1, 0)
%Y A342970 Cf. A000005, A033950, A342969.
%K A342970 nonn
%O A342970 1,1
%A A342970 _Jianing Song_, Apr 01 2021