A284203 Number of twin prime (A001097) divisors of n.

0, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 2, 0, 1, 1, 1, 1, 2, 1, 0, 1, 1, 1, 1, 1, 1, 2, 1, 0, 2, 1, 2, 1, 0, 1, 2, 1, 1, 2, 1, 1, 2, 0, 0, 1, 1, 1, 2, 1, 0, 1, 2, 1, 2, 1, 1, 2, 1, 1, 2, 0, 2, 2, 0, 1, 1, 2, 1, 1, 1, 0, 2, 1, 2, 2, 0, 1, 1, 1, 0, 2, 2, 1, 2, 1, 0, 2, 2, 0, 2, 0, 2, 1, 0, 1, 2, 1, 1, 2, 1, 1, 3, 0, 1, 1, 1, 2

Offset: 1

Author

Ilya Gutkovskiy, Mar 22 2017

Keywords

• nonn

Examples

			--------------------------------------------
| n | divisors of n | twin prime    | a(n) |
|   |               | divisors of n |      |
|------------------------------------------
| 1 | {1}           |      {-}      |  0   |
| 2 | {1, 2}        |      {-}      |  0   |
| 3 | {1, 3}        |      {3}      |  1   |
| 4 | {1, 2, 4}     |      {-}      |  0   |
| 5 | {1, 5}        |      {5}      |  1   |
| 6 | {1, 2, 3, 6}  |      {3}      |  1   |
| 7 | {1, 7}        |      {7}      |  1   |
| 8 | {1, 2, 4, 8}  |      {-}      |  0   |
| 9 | {1, 3, 9}     |      {3}      |  1   |
--------------------------------------------
		

Links

• Eric Weisstein's World of Mathematics, Twin Primes.

Crossrefs

Cf. A001097, A001221, A005087, A037074, A062729, A065421, A284599.

Cf. A048599 (positions of records).

Programs

Mathematica

nmax = 110; Rest[CoefficientList[Series[Sum[Boole[PrimeQ[k] && (PrimeQ[k - 2] || PrimeQ[k + 2])] x^k/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x]]
Table[Length[Select[Divisors[n], PrimeQ[#] && (PrimeQ[# - 2] || PrimeQ[# + 2]) &]], {n, 110}]

PARI

concat([0, 0],Vec(sum(k=1, 110, (isprime(k) && (isprime(k - 2) || isprime(k + 2)))* x^k/(1 - x^k)) + O(x^111))) \\ Indranil Ghosh, Mar 22 2017

PARI

a(n) = sumdiv(n, d, isprime(d) && (isprime(d-2) || isprime(d+2))); \\ Amiram Eldar, Jun 03 2024

Python

from sympy import isprime, divisors
print([len([i for i in divisors(n) if isprime(i) and (isprime(i - 2) or isprime(i + 2))]) for n in range(1, 111)]) # Indranil Ghosh, Mar 22 2017

Formula

G.f.: Sum_{k>=1} x^A001097(k)/(1 - x^A001097(k)).

a(A062729(n)) = 0. - Ilya Gutkovskiy, Apr 02 2017

From Amiram Eldar, Jun 03 2024: (Start)

a(A048599(n)) = n.

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = A065421 - 1/5 = 1.7021605... . (End)

Additive with a(p^e) = 1 if p is in A001097, and 0 otherwise. - Amiram Eldar, May 15 2025

a(A037074(n)) = 2. - Michel Marcus, May 15 2025