A331564 Number of twin primes p < A014574(n), such that p*A014574(n) is in A014574.

1, 2, 1, 1, 2, 3, 5, 1, 1, 0, 2, 6, 2, 2, 2, 2, 2, 4, 4, 1, 5, 4, 4, 6, 7, 9, 5, 2, 2, 7, 7, 5, 3, 6, 7, 6, 7, 7, 5, 5, 3, 4, 4, 6, 5, 7, 8, 7, 6, 4, 3, 10, 3, 3, 9, 3, 4, 3, 5, 12, 4, 7, 8, 3, 5, 7, 11, 5, 8, 17, 7, 3, 6, 6, 5, 7, 6, 19, 9, 4, 10, 12, 15, 9, 4, 9, 15, 7, 19

Offset

1,2

Comments

Conjecture: a(n) > 0 for all n, except n = 10 where A014574(n) = 108. The conjecture is true for A014574(n) < 1000000.

Links

Table of n, a(n) for n=1..89.

Example

a(1) = 1, because the first average of a twin prime pair is 4 and 4*3 = 12 is also an average of a twin prime pair.
a(2) = 2, because 6*3 = 18 and 6*5 = 30.
a(3) = 1, because 12*5 = 60.
a(4) = 1, because 18*11 = 198.
a(5) = 2, because 30*5 = 150 and 30*19 = 570.
a(10) = 0, because there are no twin primes p less than 108 such that 108*p is an average of a twin prime pair.

Mathematica

twinMidQ[n_] := And @@ PrimeQ[n + {-1, 1}]; twinMids = Select[Range[3500], twinMidQ]; numPrimes[n_] := Module[{p = 2, count = 0}, While[p < n, If[Or @@ PrimeQ[p + {-2, 2}] && twinMidQ[p * n], count++]; p = NextPrime[p] ]; count]; numPrimes /@ twinMids (* Amiram Eldar, Jan 21 2020 *)

Crossrefs

Cf. A014574.

Sequence in context: A320767 A377090 A058713 * A288250 A169786 A191631

Adjacent sequences: A331561 A331562 A331563 * A331565 A331566 A331567

Keyword

nonn

Author

Dmitry Kamenetsky, Jan 20 2020

Status

approved