

A240232


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


2



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
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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. 32=1, 2+1^2=3 is a prime number. So a(1)=1;
n=2, the second prime number is 3. 53=2, 3+2^2=7 is a prime number. So a(2)=2;
n=3, the third prime number is 5. 75=2, 5+2^2=9 is not prime number. 115=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: A065630 A319376 A110633 * A119250 A059773 A127399
Adjacent sequences: A240229 A240230 A240231 * A240233 A240234 A240235


KEYWORD

nonn,easy


AUTHOR

Lei Zhou, Apr 02 2014


STATUS

approved



