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Numbers k such that 10k - 3, 10k - 1, 10k + 1 and 10k + 3 are divisible only by primes congruent to 3 mod 4.
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%I #25 Nov 17 2024 07:16:48

%S 8,13,21,44,50,75,89,99,133,146,150,245,254,289,319,327,395,468,500,

%T 517,579,601,608,691,704,761,764,878,1011,1098,1125,1199,1266,1298,

%U 1313,1315,1414,1495,1544,1716,1723,1752,1762,1781,1844,2043,2073,2186,2281,2291,2309,2360,2444,2455,2457

%N Numbers k such that 10k - 3, 10k - 1, 10k + 1 and 10k + 3 are divisible only by primes congruent to 3 mod 4.

%C Prime terms: 13, 89, 601, 691, 761, 1723, 2281, 2309, 2693, 5437, 5821, 6199, ...

%H Harvey P. Dale, <a href="/A269354/b269354.txt">Table of n, a(n) for n = 1..1000</a>

%e 8 is a term because 10*8 - 3 = 77 = 7*11, 10*8 - 1 = 79, 10*8 + 1 = 81 = 3^4 and 10*8 + 3 = 83 are divisible only by primes congruent to 3 mod 4.

%p filter:= n ->

%p andmap(t -> numtheory:-factorset(t) mod 4 = {3},[10*n-3,10*n-1,10*n+1,10*n+3]):

%p select(filter, [$1..10000]); # _Robert Israel_, Feb 25 2016

%t pc3m4Q[n_]:=AllTrue[Flatten[FactorInteger[10 n+{-3,-1,1,3}],1][[All,1]], Mod[#,4]==3&]; Select[Range[2500],pc3m4Q] (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale_, Mar 30 2018 *)

%o (Magma) [n : n in [1..3000] | forall{d: d in PrimeDivisors(10*n-3) | d mod 4 eq 3}

%o and forall{d: d in PrimeDivisors(10*n-1) | d mod 4 eq 3}

%o and forall{d: d in PrimeDivisors(10*n+1) | d mod 4 eq 3}

%o and forall{d: d in PrimeDivisors(10*n+3) | d mod 4 eq 3}] ;

%Y Cf. A004614.

%K nonn,easy

%O 1,1

%A _Juri-Stepan Gerasimov_, Dec 23 2015