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%I #17 Aug 29 2023 09:45:24
%S 2,3,5,7,11,13,19,23,31,37,47,53,61,73,83,97,113,127,139,157,173,193,
%T 211,233,257,281,307,331,359,383,409,439,467,499,523,557,593,619,653,
%U 691,727,761,797,839,883,919,953,997,1039,1087,1129,1171,1223,1259,1307
%N a(1) = 2 and a(n+1) is the largest prime <= a(n) + n.
%C Conjecture: a(n+1) > a(n).
%C The conjecture holds for the first 2^32.5 =~ 6074001000 terms as all prime gaps up to 2^64 are known. - _Charles R Greathouse IV_, Apr 27 2023
%H Paolo Xausa, <a href="/A362527/b362527.txt">Table of n, a(n) for n = 1..10000</a>
%F For n > 5, a(n) < n*(n-1)/2. I believe a(n) > n^2/2 + o(n^1.05) asymptotically (Baker, Harman & Pintz). - _Charles R Greathouse IV_, Apr 27 2023
%e a(2) is the largest prime <= a(1) + 1 = 3. a(2) = 3.
%e a(3) is the largest prime <= a(2) + 2 = 5. a(3) = 5.
%e a(4) is the largest prime <= a(3) + 3 = 8. a(4) = 7.
%t A362527list[nmax_]:=Module[{n=2},NestList[NextPrime[#+n++,-1]&,2,nmax-1]];A362527list[100] (* _Paolo Xausa_, Aug 29 2023 *)
%o (Python)
%o from sympy import prevprime; L = [2]
%o for _ in range(55): a = prevprime(L[-1] + len(L) + 1); L.append(a)
%o print(*L, sep = ", ")
%o (PARI) first(n)=my(v=vector(n)); v[1]=2; for(k=1,n-1, v[k+1]=precprime(v[k]+k)); v \\ _Charles R Greathouse IV_, Apr 27 2023
%Y Cf. A095117, A113161.
%K nonn
%O 1,1
%A _Ya-Ping Lu_, Apr 23 2023
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