A088444 - OEIS

%I #10 May 08 2022 10:34:13

%S 1,2,3,2,5,2,7,2,3,5,11,3,13,7,3,4,17,3,19,4,3,11,23,3,5,13,9,7,29,5,

%T 31,8,11,17,5,6,37,19,13,5,41,6,43,11,5,23,47,6,7,5,17,13,53,6,5,7,19,

%U 29,59,5,61,31,7,8,5,6,67,17,23,5,71,6,73,37,5,19,7,6,79,5,9,41,83,6,5,43,29,8,89,5,7,23

%N Smallest divisor d of n such that all intervals [(k-1)*d+1:k*d] contain at least one prime, 1<=k<=n/d; a(1)=1.

%H Antti Karttunen, <a href="/A088444/b088444.txt">Table of n, a(n) for n = 1..20000</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PrimeGaps.html">Prime Gaps</a>

%H <a href="/index/Pri#gaps">Index entries for primes, gaps between</a>

%F a(n) = n iff n=1 or n is prime: a(A008578(n)) = A008578(n).

%F a(A002808(n)) < A002808(n).

%F A088447(a(n)) <= n <= A088448(a(n)).

%F a(n) = n / A088445(n).

%e n=30: a(30)>3 as [25:27] = {25,26,27} contains no prime, but

%e a(30)=5 as 3 in [1:5], 7 in [6:10], 11 in [11:15], 17 in [16:20], 23 in

%e [21:25], 29 in [26:30].

%o (PARI)

%o aicalop(d,u) = { for(k=1,u,for(i=1+((k-1)*d),k*d,if(isprime(i),break); if(i==(k*d),return(0)))); (1); }; \\ All Intervals Contain At Least One Prime.

%o A088444(n) = if(1==n,n,fordiv(n,d,if(aicalop(d,n/d),return(d))); (0)); \\ _Antti Karttunen_, May 08 2022

%Y Cf. A002808, A008578, A088445, A088447, A088448, A088449, A088450, A088451.

%K nonn,look

%O 1,2

%A _Reinhard Zumkeller_, Sep 30 2003

%E Data section extended up to a(92) by _Antti Karttunen_, May 08 2022