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 A307345 Numbers k such that every prime p <= sqrt(k) divides k*(k-1). 1
 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, 70, 85, 91, 105, 106, 120 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS 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. 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. LINKS Table of n, a(n) for n=1..32. P. Dusart, Estimates of Some Functions Over Primes without R.H., arXiv:1002.0442 [math.NT], 2010. EXAMPLE 120 is in the sequence because all primes <= sqrt(120) (namely 2,3,5,7) divide 120*119. MAPLE Res:= NULL: P:= 1: q:= 2: t:= 4: for n from 1 to 10^6 do if n = t then P:= P*q; q:= nextprime(q); t:= q^2 fi; if n*(n-1) mod P = 0 then Res:= Res, n fi od: Res; MATHEMATICA seqQ[k_] := AllTrue[Select[Range@Floor@Sqrt@k, PrimeQ], Divisible[k (k - 1), #] &]; Select[Range[120], seqQ] (* Amiram Eldar, Apr 03 2019 *) PROG (Sage) def isA307345(k): r = prime_range(isqrt(k)+1) return all([p.divides(k*(k-1)) for p in r]) print([n for n in (1..120) if isA307345(n)]) # Peter Luschny, Apr 03 2019 (PARI) isok(k) = forprime(p=1, sqrtint(k), if (k*(k-1) % p, return(0))); return(1); \\ Michel Marcus, Apr 05 2019 CROSSREFS Contains A323215. Sequence in context: A193096 A353744 A309129 * A033110 A049812 A093668 Adjacent sequences: A307342 A307343 A307344 * A307346 A307347 A307348 KEYWORD nonn,fini,full AUTHOR Robert Israel, Apr 03 2019 STATUS approved

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Last modified August 14 23:14 EDT 2024. Contains 375171 sequences. (Running on oeis4.)