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A038509 Composite numbers congruent to +-1 mod 6.

%I A038509 #89 Feb 08 2024 07:11:42
%S A038509 25,35,49,55,65,77,85,91,95,115,119,121,125,133,143,145,155,161,169,
%T A038509 175,185,187,203,205,209,215,217,221,235,245,247,253,259,265,275,287,
%U A038509 289,295,299,301,305,319,323,325,329,335,341,343,355,361,365,371,377,385
%N A038509 Composite numbers congruent to +-1 mod 6.
%C A038509 Or, composite numbers with smallest prime factor >= 5.
%C A038509 Or, nonprime numbers n such that binomial(n+3, 3) mod n == 1. - _Hieronymus Fischer_, Sep 30 2007
%C A038509 Note that the primes > 3 are congruent to +-1 mod 6.
%C A038509 This sequence differs from A067793 (composite n such that phi(n) > 2n/3) starting at 385. Numbers in this sequence but not in A067793 are 385, 455, 595, 665, 805, 1015, 1085, 1925, 2275, 2695, etc. See A069043. - _R. J. Mathar_, Jun 08 2008 and _Zak Seidov_, Nov 02 2011
%C A038509 Intersection of A002808 and A007310. - _Reinhard Zumkeller_, Jun 30 2012
%C A038509 The product (24/25) * (36/35) * (48/49) * (54/55) * (66/65) * (78/77) * (84/85) * (90/91) * ... * ((6*k)/a(n)) * ... = Pi^2/(6*sqrt(3)), where 6*k is the nearest number to a(n), with k in A067611 but not in A002822. (See A258414.) - _Dimitris Valianatos_, Mar 27 2017
%H A038509 Reinhard Zumkeller, <a href="/A038509/b038509.txt">Table of n, a(n) for n = 1..10000</a>
%F A038509 a(n) ~ 3n. - _Charles R Greathouse IV_, Nov 20 2012
%p A038509 A038509 := proc(n)
%p A038509     option remember;
%p A038509     if n = 1 then
%p A038509         25;
%p A038509     else
%p A038509         for a from procname(n-1)+1 do
%p A038509             if not isprime(a) and modp(a,6) in {1,5} then
%p A038509                 return a;
%p A038509             end if;
%p A038509         end do:
%p A038509     end if;
%p A038509 end proc:
%p A038509 seq(A038509(n),n=1..30) ; # _R. J. Mathar_, Feb 28 2020
%t A038509 Select[Range[1000], FactorInteger[#][[1,1]] >= 5 && ! PrimeQ[#] &] (* _Robert G. Wilson v_, Dec 19 2009 *)
%t A038509 With[{nn=400},Select[Rest[Complement[Range[nn],Prime[Range[ PrimePi[ nn]]]]], MemberQ[ {1,5},Mod[#,6]]&]] (* _Harvey P. Dale_, Feb 21 2013 *)
%t A038509 Select[Range[400],CompositeQ[#]&&MemberQ[{1,5},Mod[#,6]]&] (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale_, May 13 2019 *)
%o A038509 (Haskell)
%o A038509 a038509 n = a038509_list !! (n-1)
%o A038509 a038509_list = [x | x <- a002808_list, gcd x 6 == 1]
%o A038509 -- _Reinhard Zumkeller_, Aug 05 2014, Jun 30 2012
%o A038509 (PARI) is(n)=gcd(n,6)==1 && !isprime(n) && n>7 \\ _Charles R Greathouse IV_, Nov 20 2012
%Y A038509 Cf. A171993 (nonprimes of the form 3*k+-1).
%Y A038509 Cf. A000040, A133620-A133625, A133630, A133634-A133636.
%Y A038509 Cf. A133873, A133883, A133880, A133890, A133900, A133910.
%Y A038509 Cf. A069043, A067793 (composite n such that phi(n) > 2n/3).
%K A038509 nonn,nice
%O A038509 1,1
%A A038509 _Jeff Burch_
%E A038509 More terms from _Robert G. Wilson v_, Dec 19 2009
%E A038509 Entry revised by _N. J. A. Sloane_, Dec 31 2011, at the suggestion of _Gary Detlefs_