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Smallest k such that k*7^n is a sum of two successive primes.
9

%I #10 May 06 2021 22:10:44

%S 5,6,18,10,30,18,4,28,4,30,30,60,120,38,12,6,52,120,70,10,102,60,70,

%T 10,186,174,42,6,90,146,154,22,18,140,20,168,24,240,60,80,26,286,154,

%U 22,12,196,28,4,2,128,116,156,422,130,204,84,12,118,88,240,536,564,798,114

%N Smallest k such that k*7^n is a sum of two successive primes.

%C If a(n) == 0 (mod 7), then a(n+1) = a(n)/7.

%C Records: 5, 6, 18, 30, 60, 120, 186, 240, 286, 422, 536, 564, 798, 1010, 1074, 1334, 1434, 1474, 3706, 4108, 4370, 6160, ..., .

%C Corresponding prime are twin primes for n = 0, 17, 369, ..., .

%H Robert G. Wilson v, <a href="/A180134/b180134.txt">Table of n, a(n) for n = 0..400</a>

%t f[n_] := Block[{k = 1, j = 7^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 isprime, nextprime, prevprime

%o def ok(n):

%o if n <= 5: return n == 5

%o return not isprime(n//2) and n == prevprime(n//2) + nextprime(n//2)

%o def a(n):

%o k, pow7 = 1, 7**n

%o while not ok(k*pow7): k += 1

%o return k

%o print([a(n) for n in range(64)]) # _Michael S. Branicky_, May 06 2021

%Y Cf. A180130, A180131, A180132, A180133, A179975, A180135, A180136, A180137, A180138.

%K nonn

%O 0,1

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