A078571 - OEIS

%I #17 Nov 03 2019 06:46:43

%S 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,

%T 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,

%U 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

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

%C 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]

%H Amiram Eldar, <a href="/A078571/b078571.txt">Table of n, a(n) for n = 1..10000</a>

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

%e 12th twin prime pair = (A001359(12), A006512(12)) = (149,151), hence A014574(12) = 150 = 2*3*5*5, therefore a(12) = 4.

%e From _Robert G. Wilson v_, Aug 02 2010: (Start)

%e 2) 4, 6 and no others < 10^9.

%e 3) 12, 18, 30, 42, 102, 138, 282, 618, 642, 822, 1698, 1878, 2082, ...

%e 4) 60, 150, 198, 228, 348, 462, 522, 570, 858, 1062, 1230, 1278, ...

%e 5) 72, 108, 180, 270, 312, 420, 660, 828, 882, 1020, 1032, 1050, ...

%e 6) 240, 600, 810, 1320, 1488, 2088, 2340, 2970, 3300, 4158, 4272, ...

%e 7) 192, 432, 1620, 1872, 2268, 3000, 3120, 3528, 3672, 4050, 4128, ...

%e 8) 2112, 3168, 3360, 5280, 7128, 7560, 9000, 12240, 13680, 16632, ...

%e 9) 1152, 2592, 2688, 4800, 7488, 9720, 18048, 29760, 34848, 35280, ...

%e 10) 14592, 21600, 22272, 29568, 32832, 33600, 64152, 71808, 75168, ...

%e 11) 26112, 26880, 49920, 81648, 100800, 102912, 108288, 131712, ...

%e 12) 15360, 23040, 58368, 95232, 133632, 134400, 196992, 219648, ...

%e 13) 139968, 235008, 241920, 279552, 365568, 472392, 617472, 694272, ...

%e 14) 138240, 202752, 345600, 684288, 724992, 783360, 817152, 875520, ...

%e ... (End)

%t 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 *)

%Y Cf. A078570, A078572.

%Y Cf. A001359, A014574, A001222, A068525. [_Robert G. Wilson v_, Aug 02 2010]

%K nonn

%O 1,1

%A _Reinhard Zumkeller_, Nov 29 2002