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

1,15

Links

Table of n, a(n) for n=1..110.

Eric Weisstein's World of Mathematics, Twin Primes.

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)

Example

--------------------------------------------
| 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   |
--------------------------------------------

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}]

Prog

(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

Crossrefs

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

Cf. A048599 (positions of records).

Sequence in context: A282355 A199322 A351567 * A341594 A368774 A005087

Adjacent sequences: A284200 A284201 A284202 * A284204 A284205 A284206

Keyword

nonn,changed

Author

Ilya Gutkovskiy, Mar 22 2017

Status

approved