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 A307345 Numbers k such that every prime p <= sqrt(k) divides k*(k-1). 1

%I #25 Dec 06 2019 21:44:52

%S 1,2,3,4,5,6,7,8,9,10,12,13,15,16,18,19,21,22,24,25,30,31,36,40,45,46,

%T 70,85,91,105,106,120

%N Numbers k such that every prime p <= sqrt(k) divides k*(k-1).

%C If k is in the sequence, the first Chebyshev function theta(sqrt(k)) = Sum_{p <= sqrt(k)} log(p) <= 2 log(k). Now it is known that theta(x) = x + O(x/log(x)), so this can't happen if k is sufficiently large. Thus the sequence is finite.

%C For x >= 2, theta(x) >= x - 1.2323*x/log(x) (see Dusart, Theorem 5.2). Thus theta(sqrt(k)) > 2*log(k) for k >= 417. Since there are no other terms < 417, the largest term is 120.

%H P. Dusart, <a href="https://arxiv.org/abs/1002.0442">Estimates of Some Functions Over Primes without R.H.</a>, arXiv:1002.0442 [math.NT], 2010.

%e 120 is in the sequence because all primes <= sqrt(120) (namely 2,3,5,7) divide 120*119.

%p Res:= NULL:

%p P:= 1:

%p q:= 2: t:= 4:

%p for n from 1 to 10^6 do

%p if n = t then P:= P*q; q:= nextprime(q); t:= q^2 fi;

%p if n*(n-1) mod P = 0 then Res:= Res, n fi

%p od:

%p Res;

%t seqQ[k_] := AllTrue[Select[Range@Floor@Sqrt@k, PrimeQ], Divisible[k (k - 1), #] &]; Select[Range[120], seqQ] (* _Amiram Eldar_, Apr 03 2019 *)

%o (Sage)

%o def isA307345(k):

%o r = prime_range(isqrt(k)+1)

%o return all([p.divides(k*(k-1)) for p in r])

%o print([n for n in (1..120) if isA307345(n)]) # _Peter Luschny_, Apr 03 2019

%o (PARI) isok(k) = forprime(p=1, sqrtint(k), if (k*(k-1) % p, return(0))); return(1); \\ _Michel Marcus_, Apr 05 2019

%Y Contains A323215.

%K nonn,fini,full

%O 1,2

%A _Robert Israel_, Apr 03 2019

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Last modified September 14 03:39 EDT 2024. Contains 375911 sequences. (Running on oeis4.)