A348677 - OEIS

%I #64 Jun 11 2023 02:59:03

%S 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,

%T 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,

%U 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

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

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

%H Alois P. Heinz, <a href="/A348677/b348677.txt">Table of n, a(n) for n = 1..10000</a>

%H Michael P. May, <a href="https://doi.org/10.35834/2020/3202158">Properties of Higher-Order Prime Number Sequences</a>, Missouri J. Math. Sci. (2020) Vol. 32, No. 2, 158-170; and <a href="https://arxiv.org/abs/2108.04662">arXiv version</a>, arXiv:2108.04662 [math.NT], 2021.

%H Michael P. May, <a href="https://arxiv.org/abs/2112.08941">Approximating the Prime Counting Function via an Operation on a Unique Prime Number Subsequence</a>, arXiv:2112.08941 [math.GM], 2021.

%H Michael P. May, <a href="https://doi.org/10.35834/2023/3501105">Relationship Between the Prime-Counting Function and a Unique Prime Number Sequence</a>, Missouri J. Math. Sci. (2023), Vol. 35, No. 1, 105-116.

%F 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.

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

%F a(n) = A262275(n) - A151799(A262275(n)). - _Alois P. Heinz_, Jan 06 2022

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

%p b:= proc(n) option remember;

%p        `if`(isprime(n), 1+b(numtheory[pi](n)), 0)

%p     end:

%p g:= proc(n) option remember; local p; p:= g(n-1);

%p       do p:= nextprime(p);

%p          if b(p)::even then break fi

%p       od; p

%p     end: g(1):=3:

%p a:= n-> (t-> t-prevprime(t))(g(n)):

%p seq(a(n), n=1..86);  # _Alois P. Heinz_, Jan 06 2022

%t 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]

%t lowerP[n_]:=Module[{m}, m=n;While[!PrimeQ[m-1],m--]; m-1]

%t tab2=lowerP/@tab1

%t tab3=tab1-tab2

%Y Cf. A151799, A262275, A333242.

%K nonn

%O 1,2

%A _Michael P. May_, Oct 30 2021