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A241493
Primes p such that p + 4, p + 16, p + 64, p + 256 and p + 1024 are all semiprimes.
4
1627, 2917, 3583, 4603, 5581, 6367, 6379, 8263, 9697, 12517, 12763, 13339, 14197, 15289, 16339, 16993, 17539, 17737, 18199, 19267, 19531, 20023, 28057, 28879, 29587, 32647, 33427, 34033, 34537, 35353, 35617, 37039, 37087, 37657, 37663, 42337, 43093, 47533, 48049
OFFSET
1,1
COMMENTS
The constants in the definition (4, 16, 64, 256 and 1024 ) are in geometric progression.
LINKS
EXAMPLE
1627 is prime and appears in the sequence because 1627+4 = 1631 = 7*233, 1627+16 = 1643 = 31*53, 1627+64 = 1691 = 19*89, 1627+256 = 1883 = 7*269 and 1627+1024 = 2651 = 11*241, which are all semiprime.
MAPLE
with(numtheory): KD:= proc() local a, b, d, e, f, k; k:=ithprime(n); a:=bigomega(k+4); b:=bigomega(k+16); d:=bigomega(k+64); e:=bigomega(k+256); f:=bigomega(k+1024); if a=2 and b=2 and d=2 and e=2 and f=2 then RETURN (k); fi; end: seq(KD(), n=1..10000);
MATHEMATICA
KD = {}; Do[t = Prime[n]; If[PrimeOmega[t + 4] == 2 && PrimeOmega[t + 16] == 2 && PrimeOmega[t + 64] == 2 && PrimeOmega[t + 256] == 2 && PrimeOmega[t + 1024] == 2, AppendTo[KD, t]], {n, 10000}]; KD
(* For the b-file *) c = 0; Do[t = Prime[n]; If[PrimeOmega[t + 4] == 2 && PrimeOmega[t + 16] == 2 && PrimeOmega[t + 64] == 2 && PrimeOmega[t + 256] == 2 && PrimeOmega[t + 1024] == 2, c++; Print[c, " ", t]], {n, 1, 5*10^6}];
Select[Prime[Range[5000]], Union[PrimeOmega[#+{4, 16, 64, 256, 1024}]] == {2}&] (* Harvey P. Dale, Nov 28 2017 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
K. D. Bajpai, Apr 24 2014
STATUS
approved