A348677 a(n) is the difference between A262275(n) and the next lower prime.

1, 4, 4, 4, 6, 4, 2, 14, 6, 10, 12, 2, 6, 2, 4, 8, 4, 4, 6, 6, 6, 10, 4, 6, 4, 10, 2, 14, 14, 8, 10, 2, 18, 8, 8, 4, 10, 4, 8, 12, 6, 14, 2, 2, 2, 8, 12, 6, 10, 10, 12, 10, 8, 2, 2, 4, 6, 6, 16, 14, 6, 6, 2, 10, 6, 2, 8, 6, 20, 2, 8, 28, 6, 16, 2, 6, 2, 10, 6, 22, 4, 6, 4, 14, 6, 2

Offset: 1

Author

Michael P. May, Oct 30 2021

Keywords

• nonn

Comments

This sequence can be used as an alternate method of approximating the prime-counting function pi(n).

Examples

			For n = 3, a(3) = 17 - 13 = 4.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..10000
• Michael P. May, Properties of Higher-Order Prime Number Sequences, Missouri J. Math. Sci. (2020) Vol. 32, No. 2, 158-170; and arXiv version, arXiv:2108.04662 [math.NT], 2021.
• Michael P. May, Approximating the Prime Counting Function via an Operation on a Unique Prime Number Subsequence, arXiv:2112.08941 [math.GM], 2021.
• Michael P. May, Relationship Between the Prime-Counting Function and a Unique Prime Number Sequence, Missouri J. Math. Sci. (2023), Vol. 35, No. 1, 105-116.

Crossrefs

Cf. A151799, A262275, A333242.

Programs

Maple

b:= proc(n) option remember;
       `if`(isprime(n), 1+b(numtheory[pi](n)), 0)
    end:
g:= proc(n) option remember; local p; p:= g(n-1);
      do p:= nextprime(p);
         if b(p)::even then break fi
      od; p
    end: g(1):=3:
a:= n-> (t-> t-prevprime(t))(g(n)):
seq(a(n), n=1..86);  # Alois P. Heinz, Jan 06 2022

Mathematica

fQ[n_]:=If[!PrimeQ[n]||(PrimeQ[n]&&FreeQ[lst,PrimePi[n]]),AppendTo[lst,n]];k=2;lst={1};While[k<10000000,fQ@k;k++];tab1=Select[lst,PrimeQ]
lowerP[n_]:=Module[{m}, m=n;While[!PrimeQ[m-1],m--]; m-1]
tab2=lowerP/@tab1
tab3=tab1-tab2

Formula

a(n) = p_p'(n) - p_(p'(n) - 1), where p' is a prime number in the sequence A333242, p_p' is a prime number with index in A333242 (forms the prime number sequence A262275), and p_(p'(n)-1) is a prime number which is the next lower prime than those in A262275.

a(n) = A001223(A000720(A262275(n)) - 1).

a(n) = A262275(n) - A151799(A262275(n)). - Alois P. Heinz, Jan 06 2022