A348198 -id:A348198 - cp's OEIS frontend

A348199 a(n) is the least term of A326835 whose number of divisors is n.

1, 3, 9, 15, 81, 45, 729, 105, 225, 405, 59049, 495, 531441, 3645, 2025, 1155, 43046721, 3675, 387420489, 4455, 18225, 295245, 31381059609, 8085, 50625, 2657205, 65025, 40095, 22876792454961, 34425, 205891132094649, 19635, 1476225, 215233605, 455625, 62475

Offset: 1

Amiram Eldar, Oct 06 2021

nonn

First differs from A038547 at n = 12.
All the terms are odd since all the terms of A326835 are odd (as phi(1) = phi(2) = 1).
a(n) exists for any n since 3^(n-1) is a term of A326835 which has n divisors.

Cf. A038547, A326835, A328858, A348198.

seq[m_] := Module[{p = Select[Range[m], PrimeQ], s = Table[0, {m}], c, nd, ndd}, s[[p]] = 3^(p - 1); c = Length[p]; n = 1; While[c < m, nd = DivisorSigma[0, n]; If[nd <= m && s[[nd]] == 0, ndd = Length@Union[EulerPhi /@ Divisors[n]]; If[ndd == nd, c++; s[[nd]] = n]]; n++]; s]; seq[30]

a(n) <= 3^(n-1), with equality if n is prime.

a(n) >= A038547(n).

A348200 Terms of A348004 having more unitary divisors than any smaller term.

Original entry on oeis.org

1, 3, 12, 60, 660, 9240, 157080, 2984520, 68643960, 3226266120

Offset: 1

Amiram Eldar, Oct 06 2021

The corresponding numbers of unitary divisors are 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, ... (apparently, all the powers of 2).

a(11) > 7*10^10, if it exists.

			The sequence A348004 begins with 1, 3, 4, 5, 7, 8, 9, 11 and 12. The number of unitary divisors of these terms are 1, 2, 2, 2, 2, 2, 2, 2 and 4, respectively. The record values, 1, 2 and 4, occur at 1, 3 and 12, the first 3 terms of this sequence.

The unitary version of A348198.

Cf. A348002, A348004.

f[p_, e_] := p^e - 1; uphi[1] = 1; uphi[n_] := Times @@ f @@@ FactorInteger[n]; q[n_] := Length @ Union[uphi /@ (d = Select[Divisors[n], CoprimeQ[#, n/#] &])] == Length[d]; dm = 0; s = {}; Do[If[q[n], d = 2^PrimeNu[n]; If[d > dm, dm = d; AppendTo[s, n]]], {n, 1, 10^6}]; s

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,", ")));}

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A348199 a(n) is the least term of A326835 whose number of divisors is n.

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A348200 Terms of A348004 having more unitary divisors than any smaller term.

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

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