[seqfan] Re: Two Sequences

[Maximilian Hasler]

On Thu, Jul 4, 2019, 07:07 Tomasz Ordowski <tomaszordowski at gmail.com> wrote:

> Are there Wolstenholme pseudoprimes:
> composite numbers n such that n^2 | A001008(n-1) ?
> If not, then let's define weaker pseudoprimes:
> composite numbers m such that m | A001008(m-1).
> Question: Are there such numbers?
>

I didn't find any up to 10^5 - which is very small, if you consider e.g.
A125854(5) = 2001907169
(Primes p with the property that p divides the Wolstenholme number
A001008((p+1)/2).)
[BTW, These should not be called "Wolstenholme number",
name reserved for A007406 <https://oeis.org/A007406> = numerators of Sum
1/k^2.]
.
Note that "Wolstenholme pseudoprimes" can mean something else, see
A298946 <https://oeis.org/A298946> and related (A088164: Wolstenholme
primes, ...)

- Maximilian

śr., 3 lip 2019 o 23:46 M. F. Hasler <oeis at hasler.fr> napisał(a):
>
> > On Wed, Jul 3, 2019 at 5:17 PM C Boyd  wrote:
> > > For values > 29, the indices given match A006093 a(n) = prime(n) - 1.
> >
> > Indeed, it appears that A001008(p-1) is divisible by p^2, at least for
> the
> > first values.
> > I'm just working out drafts related to factorization of A001008 :
> bigomega,
> > list of factors with/without repetition, smallest & largest
> prime factors...
> > (cf. A308967 .. A308971)
> > (The latter seem to coincide with A120299 = Largest prime factor of
> > Stirling numbers of first kind A000254,
> > but those are the numerators without reduction to lowest terms, so
> maybe(?)
> > we can expect that the gpf() will eventually differ...).
> >
> > (I'm quite ignorant about known results concerning this topic, please
> feel
> > free to add comments & references.)
> >
> > - Maximilian
> >
> > In fact, ignoring the odd terms 7 & 29, it appears to be a subsequence of
> > > A006093.
> > >
> > > Does this similarity extend to your larger terms?
> > >
> > > CB
> > >
> > > On 03/07/2019, M. F. Hasler <seqfan at hasler.fr> wrote:
> > > > 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  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!)
> > > >
>