A359964 -id:A359964 - cp's OEIS frontend

A359963 Arithmetic numbers (A003601) having more divisors than all smaller arithmetic numbers.

1, 3, 6, 20, 30, 60, 168, 420, 840, 1980, 2160, 2520, 7560, 10080, 15120, 27720, 79200, 83160, 110880, 166320, 262080, 332640, 554400, 786240, 831600, 1081080, 1441440, 2162160, 2882880, 4324320, 7207200, 8648640, 10810800, 17297280, 21621600, 36756720, 43243200

Offset: 1

Amiram Eldar, Jan 20 2023

nonn

The corresponding numbers of divisors are 1, 2, 4, 6, 8, 12, 16, 24, 32, ... .

This sequence is infinite since there are arithmetic numbers with any number of divisors (see A359965).

Amiram Eldar, Table of n, a(n) for n = 1..55

Subsequence of A003601.
Cf. A000005, A359965.
Similar sequences: A002182, A335317, A359964.

seq[nmax_] := Module[{s = {}, dm = 0, d}, Do[d = DivisorSigma[0, n]; If[d > dm && Divisible[DivisorSigma[1, n], d], dm = d; AppendTo[s, n]], {n, 1, nmax}]; s]; seq[10^6]

lista(nmax) = {my(dm = 0, d); for(n = 1, nmax, d = numdiv(n); if(d > dm && sigma(n)%d == 0, dm = d; print1(n, ", "))); }

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

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Aug 05 2023

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.

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

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

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]

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,", ")));}

cp's OEIS Frontend

A359963 Arithmetic numbers (A003601) having more divisors than all smaller arithmetic numbers.

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