A240232 a(n) is the smallest positive integer such that prime(n) + a(n) and prime(n) + a(n)^2 are both prime numbers.

1, 2, 6, 4, 6, 4, 6, 12, 6, 12, 6, 4, 30, 4, 6, 6, 42, 6, 4, 30, 6, 10, 24, 12, 4, 6, 6, 24, 18, 36, 10, 6, 12, 10, 30, 30, 6, 4, 12, 54, 18, 10, 6, 6, 30, 28, 16, 4, 6, 12, 6, 12, 66, 30, 6, 18, 78, 6, 4, 30, 10, 18, 24, 6, 36, 30, 6, 16, 6, 10, 6, 30, 34, 6

Offset

1,2

Comments

Conjecture: For each prime(n) there exists at least one positive integer a(n) such that prime(n) + a(n) and prime(n) + (a(n))^2 are both primes.

Alternative representation of this conjecture: If defining a(n)=0 for those n such that no positive integer a(n) exists to make both prime(n)+a(n) and prime(n) + (a(n))^2 primes, it is conjectured that a(n) > 0 for all n >= 1.

Links

Lei Zhou, Table of n, a(n) for n = 1..10000

Example

n=1, the first prime number is 2. 3-2=1, 2+1^2=3 is a prime number. So a(1)=1;
n=2, the second prime number is 3. 5-3=2, 3+2^2=7 is a prime number. So a(2)=2;
n=3, the third prime number is 5. 7-5=2, 5+2^2=9 is not prime number. 11-5=6, 5+6^2=41 is a prime number. So a(3)=6;
...
n=13, the 13th prime is 41, 41+30=71, 41+30^2=941 are both prime numbers. For any number smaller than 30, there is not such a feature. So a(13)=30.

Mathematica

Table[p = Prime[n]; pt = p; While[pt = NextPrime[pt]; diff = pt - p; ! (PrimeQ[p + diff^2])]; diff, {n, 1, 74}]
spi[n_]:=Module[{k=1, pn=Prime[n]}, While[!PrimeQ[pn+k]||!PrimeQ[pn+k^2], k++]; k]; Array[spi, 80] (* Harvey P. Dale, Aug 10 2017 *)

Crossrefs

Cf. A000040, A240087.

Sequence in context: A364222 A319376 A110633 * A119250 A059773 A127399

Adjacent sequences: A240229 A240230 A240231 * A240233 A240234 A240235

Keyword

nonn,easy

Author

Lei Zhou, Apr 02 2014

Status

approved