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.
a(11) = 5 because 31^2 = 961, 1073, 1147, 1271, 1333, 1369 = 37^2 and all the terms are in that order in A368458.
A050216 := proc(n) option remember ; local p,pn ; if n = 0 then 2; else p := ithprime(n) ; pn := nextprime(p) ; numtheory[pi](pn^2)-numtheory[pi](p^2) ; end if; end proc: A380136 := proc(n) option remember ; if n = 1 then 1; else for a from procname(n-1)+1 do if A050216(a) > A050216(procname(n-1)) then return a ; end if; end do: end if; end proc: seq(A380136(n),n=1..30) ; # R. J. Mathar, Jan 27 2025
a(5) = a(6) = a(7) = a(8) = a(9) = 2 because prime(1) = 2 < sqrt(5 to 9) <= prime(2) = 3.
Table[1 + PrimePi[Sqrt[n-1]], {n, 100}] (* Alonso del Arte, Nov 01 2013 *)
from math import isqrt from sympy import primepi def A230774(n): return primepi(isqrt(n-1))+1 # Chai Wah Wu, Nov 04 2024
a(5)=a(6)=a(7)=a(8)=a(9)=3 because prime(1)= 2 < sqrt(5 to 9) <= prime(2) = 3.
spn[n_]:=Module[{s=Sqrt[n]},If[PrimeQ[s],s,NextPrime[s]]]; Array[spn,90] (* Harvey P. Dale, Feb 10 2019 *)
from math import isqrt from sympy import nextprime def A230775(n): return nextprime(isqrt(n-1)) # Chai Wah Wu, Nov 04 2024
a(1)=12 because between (prime(1))^2=2^2=4 and (prime(2))^2=3^2=9 there are 2 primes {5,7} which sum to 12
a(2)=83 because between (prime(2))^2=9 and (prime(3))^2=25 there are 5 primes {11,13,17,19,23} which sum to 83
a(3)=228 because between 5^2=25 and 7^2=49 there are 6 primes {29,31,37,41,43,47} which sum to 228
a(4)=1265 because between 49 and 121 there are 15 primes {53..113} which sum to 1265
a(5)=1321 because between 121 and 169 there are 9 primes {127..167} which sum to 1321.
Table[Total[Select[Range[Prime[n]^2,Prime[n+1]^2],PrimeQ]],{n,60}]
Total[Select[Range[#[[1]],#[[2]]],PrimeQ]]&/@ Partition[Prime[ Range[ 40]]^2,2,1] (* Harvey P. Dale, Jul 13 2015 *)
a(2) = 21 because there are 21 primes between prime(2)^3 = 27 and prime(3)^3 = 125.
A:= [seq(numtheory:-pi(ithprime(i)^3),i=1..30)]: A[2..-1] - A[1..-2];
For n=1, (prime(1+1)-1)^2 = 4, (prime(1)+1)^2 = 9 and we have two primes between 4 and 9, so a(1)=2.
A370796:= proc (n) local count, a, b, p: count := 0: a := (ithprime(n)+1)^2: b := (ithprime(n+1)-1)^2: p := n: while ithprime(p) <= b do if a <= ithprime(p) then count := count+1 end if: p := p+1 end do: return count end proc: A370796(1) := 2: map(A370796, [$1 .. 100]);
Table[Abs[ PrimePi[(Prime[n+1]-1)^2]- PrimePi[(Prime[n]+1)^2]],{n,72}] (* James C. McMahon, Mar 02 2024 *)
from sympy import primepi, prime, nextprime def A370796(n): return -primepi(((p:=prime(n))+1)**2)+primepi((nextprime(p)-1)**2) if n>1 else 2 # Chai Wah Wu, Mar 27 2024
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