This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
113 is a term because between prime 113 and next prime 127 there exists a square 121 and a cube 125.
b[n_] := If[IntegerQ@Sqrt@n, 0, p = NextPrime[n^3, -1]; If[Floor[Sqrt[NextPrime[p]]]^2 <= p, 0, p]]; Select[Array[b@# &, 100], # > 0 &]
127 is a term because between prime 127 and previous prime 113 there exists a square 121 and a cube 125.
b[n_] := If[IntegerQ@Sqrt@n, 0, p = NextPrime[n^3]; If[Ceiling[Sqrt[NextPrime[p,-1]]]^2 >= p, 0, p]]; Select[Array[b@# &, 1000], # > 0 &]
125 is a term because it is a cube and there are no primes between 125 and 121, its nearest square.
b[n_] := If[IntegerQ@Sqrt@n, 0, p = NextPrime[n^3]; If[Ceiling[Sqrt[NextPrime[p, -1]]]^2 >= p, 0, n^3]]; Select[Array[b@# &, 1000], # > 0 &]
n = prime(9750374) = 174689077, next prime = 174689101, mean = 174689089 = 13217^2, a prime power. The arithmetic mean of two consecutive primes is never prime, while between two consecutive primes, prime powers occur. These prime powers are in the middle of gap: p+d/2 = q^w. The prime power is most often square and very rarely occurs more than once (see A053706).
fi[x_] := FactorInteger[x] ff[x_] := Length[FactorInteger[x]] Do[s=(Prime[n]+Prime[n+1])/2; s1=ff[s]; If[Equal[s1,1],Print[{n,p=Prime[n],s,fi[s],s-p,s1}]], {n,1,10000000}]
Select[Partition[Prime[Range[25*10^5]],2,1],PrimePowerQ[Mean[#]]&][[;;,1]] (* Harvey P. Dale, Oct 15 2023 *)
121 is a term because it is a square and there are no primes between 123 and 125, its nearest cube.
b[n_] := If[IntegerQ@Sqrt@n, 0, p = NextPrime[n^3]; q = Ceiling[Sqrt[NextPrime[p, -1]]]; If[q^2 >= p, 0, q]]; Select[Array[b@# &, 1000], # > 0 &]^2
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