A078571 Total number of prime factors of the average of n-th twin prime pair.

2, 2, 3, 3, 3, 3, 4, 5, 3, 5, 3, 4, 5, 7, 4, 4, 6, 5, 3, 5, 4, 5, 7, 4, 4, 4, 6, 3, 3, 5, 6, 3, 5, 4, 5, 5, 5, 5, 4, 5, 9, 4, 4, 4, 4, 6, 5, 5, 4, 6, 5, 7, 4, 3, 4, 4, 7, 3, 5, 5, 5, 5, 3, 6, 8, 4, 5, 3, 7, 5, 6, 3, 5, 9, 3, 9, 5, 5, 5, 3, 6, 7, 7, 8, 4, 4, 6, 5, 8, 4, 4, 3, 5, 7, 5, 3, 4, 7, 5, 5, 5, 3, 4, 4, 5

Offset: 1

Author

Reinhard Zumkeller, Nov 29 2002

Keywords

• nonn

Comments

Between every twin prime pair is a composite number. This sequence looks at a characteristic of those numbers. If the number, n, is the average of a twin prime pair, p&q, then n=(p+q)/2 and p*q=n^2 -1. [Robert G. Wilson v, Aug 02 2010]

Examples

			12th twin prime pair = (A001359(12), A006512(12)) = (149,151), hence A014574(12) = 150 = 2*3*5*5, therefore a(12) = 4.
From _Robert G. Wilson v_, Aug 02 2010: (Start)
2) 4, 6 and no others < 10^9.
3) 12, 18, 30, 42, 102, 138, 282, 618, 642, 822, 1698, 1878, 2082, ...
4) 60, 150, 198, 228, 348, 462, 522, 570, 858, 1062, 1230, 1278, ...
5) 72, 108, 180, 270, 312, 420, 660, 828, 882, 1020, 1032, 1050, ...
6) 240, 600, 810, 1320, 1488, 2088, 2340, 2970, 3300, 4158, 4272, ...
7) 192, 432, 1620, 1872, 2268, 3000, 3120, 3528, 3672, 4050, 4128, ...
8) 2112, 3168, 3360, 5280, 7128, 7560, 9000, 12240, 13680, 16632, ...
9) 1152, 2592, 2688, 4800, 7488, 9720, 18048, 29760, 34848, 35280, ...
10) 14592, 21600, 22272, 29568, 32832, 33600, 64152, 71808, 75168, ...
11) 26112, 26880, 49920, 81648, 100800, 102912, 108288, 131712, ...
12) 15360, 23040, 58368, 95232, 133632, 134400, 196992, 219648, ...
13) 139968, 235008, 241920, 279552, 365568, 472392, 617472, 694272, ...
14) 138240, 202752, 345600, 684288, 724992, 783360, 817152, 875520, ...
... (End)
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A078570, A078572.

Cf. A001359, A014574, A001222, A068525. [Robert G. Wilson v, Aug 02 2010]

Programs

Mathematica

f[n_] := Plus @@ Last /@ FactorInteger@n; p = 3; lst = {}; While[p < 1000, If[ PrimeQ[p + 2], AppendTo[lst, f[p + 1]]]; p = NextPrime@p]: lst (* Robert G. Wilson v, Aug 02 2010 *)

Formula

a(n) = A001222(A014574(n)).