A039731 a(n) = max{prime(n) mod q, where prime q < prime(n) = n-th prime}.

1, 2, 2, 4, 6, 6, 8, 10, 12, 14, 18, 18, 20, 18, 24, 28, 30, 30, 34, 36, 38, 40, 42, 44, 48, 50, 48, 50, 54, 60, 64, 66, 68, 70, 72, 78, 80, 78, 84, 82, 84, 94, 96, 96, 98, 104, 110, 100, 102, 106, 112, 114, 124, 126, 126, 132, 134, 138, 132, 134, 144, 150

Offset: 2

Author

Clark Kimberling

Keywords

• nonn
• easy

Comments

Sequence lists the maxima of rows of triangle A207409. - Michel Marcus, Oct 01 2013

If there is a prime q with p/2 < q < 2p/3, then p mod q = p - q and a(n) = p - nextprime(p/2). But by a result of Nagura, there is always a prime between x and 6x/5 for x > 25, so this holds for all p > 50 and (checking 2 <= n <= 15) for all n > 1. - Charles R Greathouse IV, Jul 12 2024

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 2..10000
• Jitsuro Nagura, On the interval containing at least one prime number, Proc. Japan Acad. 28: 177-181, 1952.

Crossrefs

Cf. A000040, A151800, A207409.

Programs

Maple

a := proc(n) ithprime(n); % - nextprime(iquo(%, 2)) end: seq(a(n), n = 2..63);
# Peter Luschny, Jul 13 2024  (after Charles R Greathouse IV)

Mathematica

Map[# - NextPrime[#/2] &, Prime[Range[2, 100]]] (* Paolo Xausa, Jul 15 2024, after Charles R Greathouse IV *)

PARI

a(n)=maxp = 0; for (i = 1, n-1, mp = prime(n) % prime(i); maxp = max(mp, maxp);); maxp; \\ Michel Marcus, Oct 01 2013

PARI

P=primes(100); vector(#P,i,mx=0;for(j=1,i-1, mx=max(P[i]%P[j], mx)); mx)[2..#P] \\ Charles R Greathouse IV, Oct 01 2013

PARI

a(n) = my(p=prime(n)); p - nextprime(p/2) \\ Charles R Greathouse IV, Jul 12 2024

Formula

a(n) = p - nextprime(p/2), where p is the n-th prime, see Greathouse comment. - Charles R Greathouse IV, Jul 12 2024