Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #44 Jul 16 2024 08:09:10
%S 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,
%T 50,48,50,54,60,64,66,68,70,72,78,80,78,84,82,84,94,96,96,98,104,110,
%U 100,102,106,112,114,124,126,126,132,134,138,132,134,144,150
%N a(n) = max{prime(n) mod q, where prime q < prime(n) = n-th prime}.
%C Sequence lists the maxima of rows of triangle A207409. - _Michel Marcus_, Oct 01 2013
%C 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
%H Charles R Greathouse IV, <a href="/A039731/b039731.txt">Table of n, a(n) for n = 2..10000</a>
%H Jitsuro Nagura, <a href="https://doi.org/10.3792/pja/1195570997">On the interval containing at least one prime number</a>, Proc. Japan Acad. 28: 177-181, 1952.
%F a(n) = p - nextprime(p/2), where p is the n-th prime, see Greathouse comment. - _Charles R Greathouse IV_, Jul 12 2024
%p a := proc(n) ithprime(n); % - nextprime(iquo(%, 2)) end: seq(a(n), n = 2..63);
%p # _Peter Luschny_, Jul 13 2024 (after _Charles R Greathouse IV_)
%t Map[# - NextPrime[#/2] &, Prime[Range[2, 100]]] (* _Paolo Xausa_, Jul 15 2024, after _Charles R Greathouse IV_ *)
%o (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
%o (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
%o (PARI) a(n) = my(p=prime(n)); p - nextprime(p/2) \\ _Charles R Greathouse IV_, Jul 12 2024
%Y Cf. A000040, A151800, A207409.
%K nonn,easy
%O 2,2
%A _Clark Kimberling_