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A287615 Let r = prime(n). Then a(n) is the smallest prime p such that there is a prime q with p > q > r and p mod q = r.

%I A287615 #31 Oct 06 2020 06:54:52
%S A287615 5,13,19,29,37,47,79,101,97,103,113,131,127,137,181,199,181,227,233,
%T A287615 229,239,257,277,283,311,307,317,409,383,367,389,409,439,521,463,509,
%U A287615 491,509,613,571,541,563,577,587,619,653,677,677,709,743,787,853,743,877
%N A287615 Let r = prime(n). Then a(n) is the smallest prime p such that there is a prime q with p > q > r and p mod q = r.
%C A287615 Prime p such that p = k * q + r, r < q < p primes; k even multiples such that p is minimal.
%H A287615 Charles R Greathouse IV, <a href="/A287615/b287615.txt">Table of n, a(n) for n = 1..10000</a>
%H A287615 Ophir Spector, <a href="https://sites.google.com/site/ophirscobwebs/maths-fun">C++ code and Excel file</a>
%e A287615 a(1) = 5, as r = 2, q = 3, p = 5, is the smallest prime such that 5 = 2 mod 3.
%e A287615 a(9) = 97, as r = 23, q = 37, p = 97.  97 = 2 * 37 + 23 is smaller than 139 = 4 * 29 + 23 (A129919).
%p A287615 f:= proc(n) local p,q,r;
%p A287615   r:= ithprime(n);
%p A287615   p:= r+1;
%p A287615   do
%p A287615    p:= nextprime(p);
%p A287615    q:= max(numtheory:-factorset(p-r));
%p A287615    if q > r then return p fi
%p A287615   od:
%p A287615 end proc:
%p A287615 map(f, [$1..100]); # _Robert Israel_, Jun 05 2017
%t A287615 a[n_] := Module[{p, q, r}, r = Prime[n]; p = r+1; While[True, p = NextPrime[p]; q = Max[FactorInteger[p-r][[All, 1]]]; If[q>r, Return[p]]] ];
%t A287615 Array[a, 100] (* _Jean-François Alcover_, Oct 06 2020, after _Robert Israel_ *)
%o A287615 (PARI) findfirstTerms(n)=my(t:small=0,a:vec=[]);forprime(r=2,,forprime(p=r+2,,forprime(q=r+2,p-2,if(p%q==r,a=concat(a,[p]);if(t++==n,a[1]-=2;return(a),break(2)))))) \\ _R. J. Cano_, Jun 06 2017
%o A287615 (PARI) first(n)=my(v=vector(n),best,k=1); v[1]=5; forprime(r=3,prime(n), best=oo; forprime(q=r+2,, if(q>=best, v[k++]=best; next(2)); forstep(p=r+2*q,best,2*q, if(isprime(p), best=p; break)))); v \\ _Charles R Greathouse IV_, Jun 07 2017
%Y A287615 Cf. A129919.
%K A287615 nonn,easy
%O A287615 1,1
%A A287615 _Ophir Spector_, May 27 2017