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Smallest k such that k*6^n is a sum of two successive primes.
9

%I #11 May 06 2021 21:20:21

%S 5,2,1,1,4,12,2,1,4,3,5,8,7,34,8,11,33,26,13,9,13,90,15,40,30,5,43,9,

%T 69,38,27,79,47,9,36,6,1,92,44,51,50,16,81,21,9,50,84,14,45,59,124,

%U 215,36,6,1,20,31,35,33,46,18,3,23,114,19,41,84,14,8,35,114,19,73,14,39,68,42

%N Smallest k such that k*6^n is a sum of two successive primes.

%C If a(n) == 0 (mod 6), then a(n+1) = a(n)/6.

%C Records: 5, 12, 34, 90, 92, 124, 215, 249, 592, 601, 1099, 1282, 1406, 1589, 1700, 2688, ..., .

%C Corresponding primes are twin primes for n = 0, 1, 2, 3, 4, 7, 13, 15, 28, 69, 120, 162, 251, 257, 279 ..., .

%H Robert G. Wilson v, <a href="/A180133/b180133.txt">Table of n, a(n) for n = 0..500</a>

%t f[n_] := Block[{k = 1, j = 6^n/2}, While[ h = k*j; PrimeQ@h || NextPrime[h, -1] + NextPrime@h != 2 h, k++ ]; k]; Array[f, 80, 0]

%o (Python)

%o from sympy import nextprime, prevprime

%o def sum2succ(n): return n == prevprime(n//2) + nextprime(n//2)

%o def a(n):

%o if n == 0: return 5

%o k, pow6 = 1, 6**n

%o while not sum2succ(k*pow6): k += 1

%o return k

%o print([a(n) for n in range(77)]) # _Michael S. Branicky_, May 02 2021

%Y Cf. A180130, A180131, A180132, A180134, A179975, A180135, A180136, A180137, A180138.

%K nonn

%O 0,1

%A _Zak Seidov_ & _Robert G. Wilson v_, Aug 15 2010