A088444 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.

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, 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, 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

Offset

1,2

Links

Antti Karttunen, Table of n, a(n) for n = 1..20000

Eric Weisstein's World of Mathematics, Prime Gaps

Index entries for primes, gaps between

Formula

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

a(A002808(n)) < A002808(n).

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

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

Example

n=30: a(30)>3 as [25:27] = {25,26,27} contains no prime, but
a(30)=5 as 3 in [1:5], 7 in [6:10], 11 in [11:15], 17 in [16:20], 23 in
[21:25], 29 in [26:30].

Prog

(PARI)
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.
A088444(n) = if(1==n, n, fordiv(n, d, if(aicalop(d, n/d), return(d))); (0)); \\ Antti Karttunen, May 08 2022

Crossrefs

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

Sequence in context: A086765 A079868 A280697 * A108077 A248737 A141310

Adjacent sequences: A088441 A088442 A088443 * A088445 A088446 A088447

Keyword

nonn,look

Author

Reinhard Zumkeller, Sep 30 2003

Extensions

Data section extended up to a(92) by Antti Karttunen, May 08 2022

Status

approved