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Numbers k such that k - 53, k - 1, k + 1, k + 53 are consecutive primes.
1

%I #21 Feb 09 2025 18:09:27

%S 9401700,64312710,78563130,83494350,92978310,101520540,111105090,

%T 121631580,136765860,138330780,139027950,145673850,157008390,

%U 163050090,166418280,169288530,170473410,177920850,198963210,200765250,213504870,220428600

%N Numbers k such that k - 53, k - 1, k + 1, k + 53 are consecutive primes.

%C This sequence is a subsequence of A014574 (average of twin prime pairs), A249674 (divisible by 30) and A256753.

%C The numbers n - 53 and n + 1 belong to A204665 (p such that p + 52 is the next prime).

%C The numbers n - 53 and n - 1 belong to primes p such that p + 54 is prime.

%H Karl V. Keller, Jr., <a href="/A274042/b274042.txt">Table of n, a(n) for n = 1..10000</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/TwinPrimes.html">Twin Primes</a>

%e 9401700 is the average of the four consecutive primes 9401647, 9401699, 9401701, 9401753.

%e 64312710 is the average of the four consecutive primes 64312657, 64312709, 64312711, 64312763.

%t Select[Partition[Prime[Range[122*10^5]],4,1],Differences[#]=={52,2,52}&][[All,2]]+1 (* _Harvey P. Dale_, Mar 07 2018 *)

%o (Python)

%o from sympy import isprime,prevprime,nextprime

%o for i in range(0,250000001,6):

%o if isprime(i-1) and isprime(i+1) and prevprime(i-1) == i-53 and nextprime(i+1) == i+53: print (i,end=', ')

%Y Cf. A014574, A077800 (twin primes), A249674, A256753.

%K nonn,changed

%O 1,1

%A _Karl V. Keller, Jr._, Jun 07 2016