A283265 a(n) = 1 if n is neither 2 nor a lesser or greater twin prime (in A001097), 0 otherwise.

1, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1

Offset

1

Comments

Characteristic sequence of A171821, which is complement of 2 together with the twin primes. (After the original name of the sequence)

The recurrence for this sequence is essentially the sieve of Eratosthenes for the complement of A171821. It is possible to vary the recurrence to get the characteristic sequence of the complement of any prime constellation. The products in the recurrence are over the divisors.

Links

Antti Karttunen, Table of n, a(n) for n = 1..65539

Index entries for characteristic functions

Formula

a(n) = if n is found in A171821 then 0 else 1.

a(n) = 1 - A164292(n), for n > 2.

Recurrence:

t(1, 1) = 1;

t(2, 1) = 0;

t(n, k) = if n = k then 1 else if n > 2 then if k = 1 then (1 - (Product_{i = 2 .. n} t(n, i))*(Product_{i = 2 .. n - 2} t(n - 2, i)))*(1 - (Product_{i = 2 .. n} t (n, i))*Product_{i = 2 .. n + 2} (t(n + 2, i))) else if (mod (n, k) = 0 then t(n/k, 1) else 1) else 1.

Mathematica

(* Recurrence: *) Clear[t, n, k, i, nn]; nn = 90; t[1, 1] = 1; t[2, 1] = 0; t[n_, k_] := t[n, k] = If[n == k, 1, If[n > 2, If[k == 1, (1 - Product[t[n, i], {i, 2, n}]*Product[t[n - 2, i], {i, 2, n - 2}])*(1 - Product[t[n, i], {i, 2, n}]*Product[t[n + 2, i], {i, 2, n + 2}]), If[Mod[n, k] == 0, t[n/k, 1], 1], 1]]]; Monitor[a = Table[t[n, 1], {n, 1, nn}], n]

Prog

(PARI) a(n)=if(isprime(n) && (isprime(n+2) || isprime(n-2)), 0, n!=2) \\ Charles R Greathouse IV, Mar 05 2017

Crossrefs

Cf. A164292, A171821, A001097, A129950.

Sequence in context: A373481 A174889 A005171 * A365410 A373549 A181406

Adjacent sequences: A283262 A283263 A283264 * A283266 A283267 A283268

Keyword

nonn

Author

Mats Granvik, Mar 04 2017

Extensions

More terms and new name from Antti Karttunen, Jan 03 2019

Status

approved