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.

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

Offset

0,2

Comments

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

Links

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

Formula

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

Example

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

Crossrefs

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

Sequence in context: A288967 A361106 A009094 * A236067 A134821 A229669

Adjacent sequences: A061698 A061699 A061700 * A061702 A061703 A061704

Keyword

nonn

Author

Labos Elemer, Jun 18 2001

Extensions

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

Status

approved