[seqfan] Re: Two Sequences

[Tomasz Ordowski]

Dear Maximilian,

If p > 3 is a prime, then p^2 | A001008(p-1), by Wolstenholme's theorem.
Cf. https://oeis.org/A001008 (see the second comment).
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?

Best regards,

Thomas


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 A1008(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 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!)
> > >>
> > >>
> > >
> > > --
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> > >
> >
> > --
> > Seqfan Mailing list - http://list.seqfan.eu/
> >
>
>
> --
> Maximilian
>
> --
> Seqfan Mailing list - http://list.seqfan.eu/
>