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%I #6 Aug 05 2023 13:16:47
%S 12,24,84,120,672,24384,43065,78975,81081,261261,523776,9124731,
%T 13398021,69087249,91963648,459818240,39142675143,51001180160
%N Admirable numbers with more divisors than any smaller admirable number.
%C The corresponding numbers of divisors are 6, 8, 12, 16, 24, 28, 32, 36, 40, 48, 80, 90, 96, 120, 144, 288, 360, 480, ... .
%C If there are infinitely many even perfect numbers (A000396), then this sequence is infinite, because if p is a Mersenne prime exponent (A000043) and q is an odd prime that does not divide 2^p-1, then 2^(p-1)*(2^p-1)*q is an admirable number with 4*p divisors (see A165772).
%C a(19) > 10^11.
%t admQ[n_] := (ab = DivisorSigma[1, n] - 2 n) > 0 && EvenQ[ab] && ab/2 < n && Divisible[n, ab/2];
%t seq[kmax_] := Module[{s = {}, dm = 0, d1}, Do[d1 = DivisorSigma[0, k]; If[d1 > dm && admQ[k], dm = d1; AppendTo[s, k]], {k, 1, kmax}]; s]; seq[10^6]
%o (PARI) isadm(n) = {my(ab=sigma(n)-2*n); ab>0 && ab%2 == 0 && ab/2 < n && n%(ab/2) == 0;}
%o lista(kmax) = {my(dm = 0, d1); for(k = 1, kmax, d1 = numdiv(k); if(d1 > dm && isadm(k), dm = d1; print1(k,", ")));}
%Y Cf. A000005, A000043, A000396, A109745, A111592, A165772.
%Y Similar sequences: A002182, A136404, A335008, A335317, A348198, A359963, A359964.
%K nonn,more
%O 1,1
%A _Amiram Eldar_, Aug 05 2023
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