cp's OEIS Frontend

A038606 Least k such that k-th prime > n * k.

%I A038606 #27 Apr 03 2023 10:36:09
%S A038606 1,5,12,31,69,181,443,1052,2701,6455,15928,40073,100362,251707,637235,
%T A038606 1617175,4124437,10553415,27066974,69709680,179992909,465769803,
%U A038606 1208198526,3140421716,8179002096,21338685407,55762149030,145935689361,382465573483,1003652347100
%N A038606 Least k such that k-th prime > n * k.
%C A038606 Log(a(n)) =~ -1.295 + 0.964312n. - _Robert G. Wilson v_, Jan 25 2002
%C A038606 Numbers n such that prime(n) (mod n) begins the next cycle of terms in A004648. Generally prime(i) (mod i) exceeds prime(i-1) (mod i-1) but there are numerous times where for a short run prime(i) (mod i) is minimally less than its predecessor. Here n is substantially less. See Labos's graph.
%C A038606 A090973(a(n)) = n+1. [From _Reinhard Zumkeller_, Aug 16 2009]
%C A038606 With offset 2: Index j of prime p(j) such that ceiling[p(j)/j]=n is first satisfied. a(n) = A062742(n) = A038624(n) for n >= 3. [From _Jaroslav Krizek_, Dec 13 2009]
%H A038606 Giovanni Resta, <a href="/A038606/b038606.txt">Table of n, a(n) for n = 1..50</a>
%H A038606 Labos, E. <a href="/A004648/a004648.gif">Graph of first 50000 terms of A004648</a>
%H A038606 Andrew R. Booker, <a href="https://t5k.org/nthprime/">The Nth Prime Page</a>
%F A038606 a(n) = pi(A038607(n)) = A000720(A038607(n)).
%p A038606 A038606 := proc(n)
%p A038606     for k from 1 do
%p A038606         if ithprime(k)> n*k then
%p A038606             return k;
%p A038606         end if;
%p A038606     end do:
%p A038606 end proc: # _R. J. Mathar_, Aug 24 2013
%t A038606 k = 1; Do[ While[ Floor[ Prime[k]/k] < n, k++ ]; Print[k]; k++, {n, 1, 30} ]
%o A038606 (PARI) k=1;n=1;forprime(p=3,4e9,if(p/n++>k,print1(n", ");k++)) \\ _Charles R Greathouse IV_, Sep 06 2011
%Y A038606 Cf. A038607, A004648, A038625.
%K A038606 nonn
%O A038606 1,2
%A A038606 Vasiliy Danilov (danilovv(AT)usa.net) 1998 Jul
%E A038606 Edited by _Robert G. Wilson v_, Jan 25 2002
%E A038606 a(21)=179992909 corrected by _Ray Chandler_, Dec 01 2004
%E A038606 a(29)-a(30) from _Charles R Greathouse IV_, Sep 06 2011
%E A038606 a(31)-a(50) obtained from the values of A038625 computed by _Jan Büthe_. - _Giovanni Resta_, Sep 01 2018