A167511 -id:A167511 - cp's OEIS frontend

A167529 a(n) is the number of nonisolated nonprimes k such that (n-th nonisolated prime) < k < (n-th isolated prime).

0, 9, 19, 22, 27, 34, 42, 37, 42, 41, 50, 50, 53, 64, 69, 49, 54, 79, 90, 72, 86, 82, 87, 74, 86, 90, 96, 106, 111, 98, 103, 102, 107, 88, 91, 88, 95, 73, 80, 73, 76, 22, 29, 26, 37, 21, 24, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Juri-Stepan Gerasimov, Nov 06 2009

nonn

			a(1)=0 (3 > [none] > 2);
a(2)=9 (5 < 8,9,10,14,15,16,20,21,22 < 23);
a(3)=19 (7 < 8,9,10,14,15,16,20,21,22,24,25,26,27,28,32,33,34,35,36 < 37).

Cf. A001097 (the nonisolated primes), A007510 (the isolated primes), A164276 (the nonisolated nonprimes), A167511.

isA001097 := proc(n) isprime(n) and ( isprime(n+2) or isprime(n-2) ); end proc:
A001097 := proc(n) option remember; if n =1 then 3; else for a from procname(n-1)+2 by 2 do if isA001097(a) then return a; end if; end do: end if; end proc:
A007510 := proc(n) option remember; if n <= 2 then op(n,[2,23]) ; else for a from procname(n-1)+2 by 2 do if isprime(a) and not isprime(a+2) and not isprime(a-2) then return a; end if; end do: end if; end proc:
A167529 := proc(n) a := 0 ; for k from A001097(n)+1 to A007510(n)-1 do if isA164276(k) then a := a+1 ; end if; end do: a ; end proc:
seq(A167529(n),n=1..120) ; # R. J. Mathar, May 30 2010

Corrected (23 replaced with 22, 28 with 27) and extended by R. J. Mathar, May 30 2010

A167595 The number of nonisolated primes between the n-th nonisolated nonprime and the n-th isolated nonprime.

Original entry on oeis.org

1, 2, 1, 3, 5, 5, 7, 9, 9, 11, 13, 15, 17, 19, 21, 23, 23, 25, 27, 29, 31, 33, 35, 37, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 61, 63, 65, 67, 69, 71, 73, 75, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99, 101, 103, 105, 107, 109, 111, 113, 115, 117, 119, 119

Offset: 1

Juri-Stepan Gerasimov, Nov 06 2009

			a(1)=1  (0 < 3 < 4);
a(2)=2  (1 < 3,5 < 6);
a(3)=1  (8 < 11 <12);
a(4)=3  (9 < 11,13,17 < 18);
a(5)=5 (10 < 11,13,17,19,29 < 30);
a(6)=5 (14 < 17,19,29,31,41 < 42);
a(7)=7 (15 < 17,19,29,31,41,43,59 < 60);
a(8)=9 (16 < 17,19,29,31,41,43,59,61,71 < 72).

Cf. A001097 (the nonisolated primes), A014574 (the isolated nonprimes), A164276 (the nonisolated nonprimes), A167511.

A164276 := proc(n) option remember; if n = 1 then 0; else for a from procname(n-1)+1 do if isA164276(a) then return a; end if; end do; end if: end proc:
A001097 := proc(n) option remember; if n =1 then 3; else for a from procname(n-1)+2 by 2 do if isA001097(a) then return a; end if; end do: end if; end proc:
A167595 := proc(n) a := 0 ; for k from A164276(n)+1 to A014574(n)-1 do if isA001097(k) then a := a+1 ; end if; end do: a ; end proc:
seq(A167595(n),n=1..120) ; # R. J. Mathar, May 30 2010

More terms from R. J. Mathar, May 30 2010

A167596 The number of isolated nonprimes between the nonisolated prime and the isolated prime.

Original entry on oeis.org

0, 3, 3, 4, 3, 4, 4, 4, 3, 3, 4, 4, 3, 5, 4, 4, 3, 6, 5, 6, 6, 6, 5, 6, 6, 6, 6, 6, 5, 6, 5, 5, 4, 4, 3, 3, 2, 2, 1, 3, 2, 2, 1, 2, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Juri-Stepan Gerasimov, Nov 06 2009

nonn

			a(1)=0 (3>no>2); a(2)=3 (5<6&12&18<23); a(3)=3 (7<12718&30<37); a(4)=4 (11<12&18&30&42<47); a(5)=3 (13<18&30&42<53); a(6)=4 (17<18&30&42&60<67).

Cf. A001097 (the nonisolated primes), A007510 (the isolated primes), A014574 (the isolated nonprimes), A167511.

All numbers from a(49) onwards corrected and sequence extended beyond a(59) by R. J. Mathar, May 30 2010

cp's OEIS Frontend

A167529 a(n) is the number of nonisolated nonprimes k such that (n-th nonisolated prime) < k < (n-th isolated prime).

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A167595 The number of nonisolated primes between the n-th nonisolated nonprime and the n-th isolated nonprime.

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A167596 The number of isolated nonprimes between the nonisolated prime and the isolated prime.

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