A362527 a(1) = 2 and a(n+1) is the largest prime <= a(n) + n.

2, 3, 5, 7, 11, 13, 19, 23, 31, 37, 47, 53, 61, 73, 83, 97, 113, 127, 139, 157, 173, 193, 211, 233, 257, 281, 307, 331, 359, 383, 409, 439, 467, 499, 523, 557, 593, 619, 653, 691, 727, 761, 797, 839, 883, 919, 953, 997, 1039, 1087, 1129, 1171, 1223, 1259, 1307

Offset: 1

Author

Ya-Ping Lu, Apr 23 2023

Keywords

• nonn

Comments

Conjecture: a(n+1) > a(n).

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

Examples

			a(2) is the largest prime <= a(1) + 1 = 3. a(2) = 3.
a(3) is the largest prime <= a(2) + 2 = 5. a(3) = 5.
a(4) is the largest prime <= a(3) + 3 = 8. a(4) = 7.
		

Links

• Paolo Xausa, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A095117, A113161.

Programs

Mathematica

A362527list[nmax_]:=Module[{n=2},NestList[NextPrime[#+n++,-1]&,2,nmax-1]];A362527list[100] (* Paolo Xausa, Aug 29 2023 *)

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

Python

from sympy import prevprime; L = [2]
for _ in range(55): a = prevprime(L[-1] + len(L) + 1); L.append(a)
print(*L, sep = ", ")

Formula

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