A359963 -id:A359963 - cp's OEIS frontend

A359964 Refactorable numbers (A033950) having more divisors than all smaller refactorable numbers.

1, 2, 8, 12, 24, 36, 60, 180, 240, 360, 720, 1260, 1680, 3360, 5040, 10080, 15120, 20160, 25200, 30240, 55440, 100800, 110880, 221760, 277200, 443520, 665280, 720720, 1108800, 1441440, 2494800, 2882880, 3603600, 5765760, 8648640, 12972960, 14414400, 25945920, 28828800

Offset: 1

Amiram Eldar, Jan 20 2023

nonn

The corresponding numbers of divisors are 1, 2, 4, 6, 8, 9, 12, 18, 20, 24, ... .

This sequence if infinite since there are refactorable numbers with arbitrarily large number of divisors. E.g., for any prime p, p^(p-1) is a refactorable number with p divisors.

Amiram Eldar, Table of n, a(n) for n = 1..60

Subsequence of A033950.
Cf. A000005, A073904, A110821.
Similar sequences: A002182, A335317, A356078, A359963.

seq[nmax_] := Module[{s = {}, dm = 0, d}, Do[d = DivisorSigma[0, n]; If[d > dm && Divisible[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 && n%d == 0, dm = d; print1(n, ", "))); }

A359965 a(n) is the least arithmetic number (A003601) having exactly n divisors.

Original entry on oeis.org

1, 3, 49, 6, 14641, 20, 594823321, 30, 8281, 304, 41426511213649, 60, 491258904256726154641, 832, 717409, 168, 160470643909878751793805444097921, 612, 114445997944945591651333831028437092270721, 432, 87616, 44032, 6111571184724799803076702357055363809, 420, 13521270961

Offset: 1

Amiram Eldar, Jan 20 2023

nonn

a(n) is the least number k such that A000005(k) = n and n | A000203(k).

a(n) exists for all n: for example, if p is a prime such that p == 1 (mod n), then p^(n-1) has n divisors and n | A000203(p^(n-1)).

Subsequence of A003601.
Cf. A000005, A000203, A359963.
Similar sequences: A005179, A073904.

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

A359964 Refactorable numbers (A033950) having more divisors than all smaller refactorable numbers.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

A359965 a(n) is the least arithmetic number (A003601) having exactly n divisors.

Views

Author

Keywords

Comments

Crossrefs

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

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

PARI