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

Author

Lei Zhou, Apr 02 2014

Keywords

• nonn
• easy

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.

Examples

			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.
		

Links

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

Crossrefs

Cf. A000040, A240087.

Programs

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 *)