A269354 Numbers n such that 10n - 3, 10n - 1, 10n + 1 and 10n + 3 are divisible only by primes congruent to 3 mod 4.

8, 13, 21, 44, 50, 75, 89, 99, 133, 146, 150, 245, 254, 289, 319, 327, 395, 468, 500, 517, 579, 601, 608, 691, 704, 761, 764, 878, 1011, 1098, 1125, 1199, 1266, 1298, 1313, 1315, 1414, 1495, 1544, 1716, 1723, 1752, 1762, 1781, 1844, 2043, 2073, 2186, 2281, 2291, 2309, 2360, 2444, 2455, 2457

Offset

1,1

Comments

Primes: 13, 89, 601, 691, 761, 1723, 2281, 2309, 2693, 5437, 5821, 6199, ...

Links

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Example

8 is in this sequence 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 primes congruent to 3 mod 4.

Maple

filter:= n ->
   andmap(t -> numtheory:-factorset(t) mod 4 = {3}, [10*n-3, 10*n-1, 10*n+1, 10*n+3]):
select(filter, [$1..10000]); # Robert Israel, Feb 25 2016

Mathematica

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 *)

Prog

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

Crossrefs

Cf. A004614.

Sequence in context: A063849 A273980 A101642 * A195984 A019535 A229446

Adjacent sequences: A269351 A269352 A269353 * A269355 A269356 A269357

Keyword

nonn,easy

Author

Juri-Stepan Gerasimov, Dec 23 2015

Status

approved