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A348677
a(n) is the difference between A262275(n) and the next lower prime.
3
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
1,2
COMMENTS
This sequence can be used as an alternate method of approximating the prime-counting function pi(n).
LINKS
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, Relationship Between the Prime-Counting Function and a Unique Prime Number Sequence, Missouri J. Math. Sci. (2023), Vol. 35, No. 1, 105-116.
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
EXAMPLE
For n = 3, a(3) = 17 - 13 = 4.
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
CROSSREFS
KEYWORD
nonn
AUTHOR
Michael P. May, Oct 30 2021
STATUS
approved