A240697 The smallest p2-p1 when 2n = p1+p4 = p2+p3, where p1 < p2 <= p3 < p4 are prime numbers.

0, 0, 0, 0, 2, 0, 4, 2, 2, 4, 2, 2, 4, 6, 2, 10, 2, 2, 12, 6, 2, 4, 2, 2, 4, 6, 2, 6, 6, 2, 12, 2, 2, 24, 6, 2, 4, 2, 2, 6, 6, 2, 4, 12, 2, 6, 6, 4, 6, 6, 2, 4, 2, 2, 4, 2, 2, 4, 6, 2, 6, 6, 2, 12, 6, 2, 4, 6, 2, 6, 2, 2, 6, 6, 2, 10, 2, 2, 12, 6, 2, 6, 6, 2, 4

Offset

1,5

Comments

a(n) is defined as 0 when 2n cannot be written as the sum of two prime numbers in two or more distinct ways.

Note that there are only two values of this sequence that are greater than 12 for n<=10000, which are a(34)=24 and a(229)=18.

From testing up to n=300000, it appears that only limited terms take a value other than 2, 4, or 6. Here is a list:

a(n) = 0 for n = 1, 2, 3, 4, and 6;

a(n) = 10 for n = 16, 76, and 181;

a(n) = 12 for n = 19, 31, 44, 64, 79, 86, 124, 139, 259, 269, 274, 314, 391, 404, 484, 499, 604, 619, 691, 709, 859, 1054, 1129, 1399, 1469, 1559, 1574, 2729, 4006, and 5804;

a(229) = 18; and a(34) = 24.

Conjecture: except for those listed above, no terms in this sequence take a value other than 2, 4, or 6.

Links

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

Example

When n<5, 2n cannot be written as the sum of two prime numbers in two distinct ways, so a(1) = a(2) = a(3) = a(4) = 0.
When n=5, 2n=10 = 3+7 = 5+5, 5-3=2, so a(5)=2.
...
When n=18, 2n=36 = 5+31 = 7+29 = 13+23 = 17+19. The smallest difference among the four numbers {5,7,13,17} is 2 = 7-5. So a(18)=2.

Mathematica

Table[p = 2; While[! PrimeQ[2*n - p], p = NextPrime[p]]; p1 = p; c = 2*n; While[p1 = NextPrime[p1]; (2*p1) <= 2*n, If[PrimeQ[2*n - p1], c = Min[c, p1 - p]; p = p1]]; If[c == 2*n, c = 0]; c, {n, 1, 85}]

Crossrefs

Cf. A000040, A002375.

Sequence in context: A274706 A037035 A159984 * A271230 A112824 A195133

Adjacent sequences: A240694 A240695 A240696 * A240698 A240699 A240700

Keyword

nonn,easy

Author

Lei Zhou, Apr 10 2014

Status

approved