A076472 -id:A076472 - cp's OEIS frontend

A076471 Number of pairs (p,q) of successive primes with p+q<=n.

0, 0, 0, 0, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 13

Offset: 1

Reinhard Zumkeller, Oct 14 2002

nonn

			Pairs (p,q) of successive primes with p+q<=27: {(2,3), (3,5), (5,7), (7,11), (11,13)}, hence a(27)=5.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000720, A056172, A076472, A076473.

f:= proc(n) local p;
  if n <= 4 then return 0 fi;
  p:= prevprime(ceil(n/2));
  if p + nextprime(p) <= n then numtheory:-pi(p) else numtheory:-pi(p)-1 fi
end proc:
map(f, [$1..100]); # Robert Israel, Dec 08 2024

With[{t=Total/@Partition[Prime[Range[100]],2,1]},Table[Count[t,?(#<=n&)],{n,100}]] (* _Harvey P. Dale, Apr 16 2015 *)

A056172(n)-1 <= a(n) <= A056172(n).

a(n) = A076472(n) + A076473(n).

A076473 Number of pairs (p,q) of successive primes with p+q<=n and gcd(p+q,n)=1.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 1, 2, 0, 2, 1, 3, 1, 1, 1, 3, 1, 4, 0, 2, 1, 4, 1, 4, 1, 2, 1, 5, 0, 6, 1, 2, 1, 4, 1, 7, 1, 2, 0, 7, 1, 8, 1, 1, 1, 8, 1, 7, 0, 2, 1, 9, 1, 7, 1, 3, 1, 9, 0, 10, 1, 3, 1, 6, 1, 10, 1, 4, 0, 11, 1, 11, 1, 3, 1, 10, 1, 12, 0, 4, 1, 12, 1, 9, 1, 4, 1, 13, 0, 10, 1, 4, 1, 10, 1, 14, 1, 4

Offset: 1

Reinhard Zumkeller, Oct 14 2002

			n=27: gcd(2+3,27)=1, gcd(3+5,27)=1, gcd(5+7,27)=3, gcd(7+11,27)=9, gcd(11+13,27)=3, hence a(27)=2.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000010, A076471, A076472.

f:= proc(n) local t,p,q,s;
  p:= 2; t:= 0;
  do
    q:= p; p:= nextprime(p);
    s:= q+p;
    if s > n then return t fi;
    if igcd(s,n) = 1 then t:= t+1 fi
  od
end proc:
map(f, [$1..100]); # Robert Israel, Dec 08 2024

a(n) = A076471(n) - A076472(n).

cp's OEIS Frontend

A076471 Number of pairs (p,q) of successive primes with p+q<=n.

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