[seqfan] Re: Two Sequences

[M. F. Hasler]

PS: it appears that indices of non-squarefree terms of
A001008 Numerators of harmonic numbers H(n) = Sum_{i=1..n} 1/i.
are:
4,6,7,10,12,16,18,22,28,29,30,36,40,42,46,52,58,60,66,70,72,78,82,88,96,
100,102,106,108,112,126,130,136,138,148,150,
156,162,166,172,178,180,190,192,196,198,
210,222,226,228,232,238,240,250,256,262,268,270,276,...
(sequence not in the OEIS -- could it be worth adding ?)

It appears that
A1008(848) = 11^3 * 1871 * C359
is the first term to have a cubic factor.

(I say "it appears" because I did not factor large cofactors.)

Could anyone confirm and / or know whether this is studied anywhere in the
literature ?

- Maximilian


On Wed, Jul 3, 2019 at 2:35 PM M. F. Hasler <seqfan at hasler.fr> wrote:

> On Wed, Jul 3, 2019 at 4:19 AM Ali Sada via SeqFan <seqfan at list.seqfan.eu>
> wrote:
>
>> The first sequence is 1, 66, 6, 12, 8220, 20, 420,213080, 17965080,
>> 153720, 210951720, 14109480, 31766925960,....
>> A(n) is the least integer that when multiplied by the harmonic number sum
>> of n generates a square number (M*M.)
>> The second sequence is M itself.
>> 1, 3, 11, 5, 137, 7, 33, 761, 7129, 671, 83771, 6617,....
>> I searched OEIS and found that this sequence is related to A120299
>> (Largest prime factor of Stirling numbers of first kind s(n,2) A000254.)
>> A120299 also represents the largest prime factor of M, except for M(1.)
>>
>
> A(n,H=sum(k=1,n,1/k))=core(numerator(H))*denominator(H)}
> /*=A007913(A001008(n))*A002805(n)*/
> A_vec(13)
> %21 = [1, 6, 66, 12, 8220, 20, 420, 213080, 17965080, 153720, 2320468920,
> 14109480, 412970037480]
> Then, M = sqrt(H*A) = sqrt(N*core(N)) = A019554(N) (= "outer square root")
> of N = numerator(H),
> this does not depend at all on the denominator, and it could be simpler to
> use  A(n) = M(n)^2/H(n) .)
>
> M_vec(Nmax, H=0)=vector(Nmax, k, H+=1/k; sqrtint(core(N=numerator(H))*N))
>  = [1, 3, 11, 5, 137, 7, 33, 761, 7129, 671, 83711, 6617, 1145993]
>
> (So far this yields the same result as the radical of N, A007947, but this
> will no more be the case as soon as N has a cube as factor. However, this
> isn't the case soon - you'll have to go beyond 150 terms or more!)
>
>