A061680 -id:A061680 - cp's OEIS frontend

A069785 a(n) = A061680(n!).

1, 1, 1, 1, 1, 15, 15, 3, 5, 135, 135, 99, 99, 9, 63, 21, 21, 459, 459, 135, 19, 15, 15, 15, 21, 189, 189, 585, 585, 18225, 18225, 675, 15, 135, 891, 8505, 25515, 81, 81, 7695, 7695, 1575, 1575, 4725, 6615, 40635, 40635, 945, 1215, 3645, 3645, 151875, 151875

Offset: 1

Labos Elemer, Apr 09 2002

			Observe cases when consecutive terms are equal: n={1,2,3,4,6,10,...,78,80,82,88,96}.

Amiram Eldar, Table of n, a(n) for n = 1..1000

Cf. A000005, A000142, A061680, A061701, A048691, A069780, A069781, A069782, A069783, A069784.

a[n_] := Module[{e = FactorInteger[n!][[;;, 2]]}, GCD[Times @@ (2*e+1), Times @@ (e+1)]]; Array[a, 100] (* Amiram Eldar, Dec 02 2023 *)

a(n) = {my(e = factor(n!)[,2]); gcd(vecprod(apply(x -> 2*x+1, e)), vecprod(apply(x -> x+1, e)));} \\ Amiram Eldar, Dec 02 2023

a(n) = A061680(A000142(n)). - Amiram Eldar, Dec 02 2023

A061701 Smallest number m such that GCD of d(m^2) and d(m) is 2n+1 where d(m) is the number of divisors of m.

Original entry on oeis.org

1, 12, 4608, 1728, 1260, 509607936, 2985984, 144, 56358560858112, 5159780352, 302400, 6232805962420322304, 207360000, 887040, 201226394483583074212773888, 15407021574586368, 248832, 2286144000, 26623333280885243904, 522547200, 8430527379596857675529996470321152

Offset: 0

Labos Elemer, Jun 18 2001

nonn

a(n) exists for every n. In other words, every positive odd integer k is equal to the GCD of d(m^2) and d(m) for some m. To see this, let m = 2^(k^2 - 1) * 3^((k-1)/2). Then d(m) = k^2 * (k+1)/2 and d(m^2) = (2 k^2 - 1) * k. Both of these are divisible by k and (8k-4) d(m) - (2k+1) d(m^2) = k, so the GCD is k. - Dean Hickerson, Jun 23 2001

All the terms are in A025487 because A061680(m) = gcd(d(m^2), d(m)) depends only on the prime signature of m. - Amiram Eldar, Nov 26 2023

			For n = 7, GCD[d(20736),d(144)] = GCD[45,15] = 15 = 2*7+1.

Amiram Eldar, Table of n, a(n) for n = 0..28

Cf. A000005, A000290, A025487, A048691, A061680.

a(n) = Min[m : GCD[d(m^2), d(m)] = 2n+1].

More terms from David Wasserman, Jun 20 2002

a(12)-a(13) corrected and a(17)-a(20) added by Amiram Eldar, Nov 26 2023

cp's OEIS Frontend

A069785 a(n) = A061680(n!).

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