A364726 Admirable numbers with more divisors than any smaller admirable number.

12, 24, 84, 120, 672, 24384, 43065, 78975, 81081, 261261, 523776, 9124731, 13398021, 69087249, 91963648, 459818240, 39142675143, 51001180160

Offset

1,1

Comments

The corresponding numbers of divisors are 6, 8, 12, 16, 24, 28, 32, 36, 40, 48, 80, 90, 96, 120, 144, 288, 360, 480, ... .

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).

a(19) > 10^11.

Links

Table of n, a(n) for n=1..18.

Mathematica

admQ[n_] := (ab = DivisorSigma[1, n] - 2 n) > 0 && EvenQ[ab] && ab/2 < n && Divisible[n, ab/2];
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]

Prog

(PARI) isadm(n) = {my(ab=sigma(n)-2*n); ab>0 && ab%2 == 0 && ab/2 < n && n%(ab/2) == 0; }
lista(kmax) = {my(dm = 0, d1); for(k = 1, kmax, d1 = numdiv(k); if(d1 > dm && isadm(k), dm = d1; print1(k, ", "))); }

Crossrefs

Cf. A000005, A000043, A000396, A109745, A111592, A165772.

Similar sequences: A002182, A136404, A335008, A335317, A348198, A359963, A359964.

Sequence in context: A120360 A120356 A109745 * A226493 A371922 A051385

Adjacent sequences: A364723 A364724 A364725 * A364727 A364728 A364729

Keyword

nonn,more

Author

Amiram Eldar, Aug 05 2023

Status

approved