A240087 -id:A240087 - cp's OEIS frontend

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

Lei Zhou, Apr 02 2014

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.

			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.

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

Cf. A000040, A240087.

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 *)

A240233 a(n) is the smallest prime number such that both a(n) + 6n and a(n) + 12n are prime numbers.

Original entry on oeis.org

5, 5, 5, 5, 7, 7, 5, 5, 5, 7, 5, 7, 11, 5, 11, 5, 7, 23, 13, 11, 5, 5, 41, 5, 7, 37, 29, 11, 5, 13, 7, 5, 13, 23, 13, 7, 5, 5, 23, 11, 11, 5, 5, 13, 7, 5, 29, 23, 13, 7, 5, 19, 41, 13, 17, 11, 7, 5, 19, 7, 7, 7, 5, 5, 7, 5, 7, 11, 29, 13, 5, 17, 5, 19, 7, 7, 5

Offset: 1

Lei Zhou, Apr 02 2014

a(n), a(n) + 6n, and a(n) + 12n form an arithmetic progression with a common difference of 6n.
If the interval is not a multiple of six, such an arithmetic progression of primes cannot exist unless a(n)=3. For example, 3,5,7 has an interval of 2; 3,7,11 has an interval of 4; and 3,11,19 has an interval of 8, as in A115334 and A206037.
Conjecture: a(n) is defined for all n > 0.

			n=1, 6n=6. 5,11,17 are all prime numbers with an interval of 6. So a(1)=5;
...
n=13, 6n=78. 5+78=83, 5+2*78=161=7*23(x); 7+78=85(x); 11+78=89, 11+78*2=167. 11,89,167 are all prime numbers with an interval of 78. So a(13)=11.

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

Cf. A000040, A115334, A206037, A240087.

Table[diff = n*6; k = 1; While[k++; p = Prime[k]; cp1 = p + diff; cp2 = p + 2*diff; ! ((PrimeQ[cp1]) && (PrimeQ[cp2]))]; p, {n, 77}]

cp's OEIS Frontend

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

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