A038509 Composite numbers congruent to +-1 mod 6.

25, 35, 49, 55, 65, 77, 85, 91, 95, 115, 119, 121, 125, 133, 143, 145, 155, 161, 169, 175, 185, 187, 203, 205, 209, 215, 217, 221, 235, 245, 247, 253, 259, 265, 275, 287, 289, 295, 299, 301, 305, 319, 323, 325, 329, 335, 341, 343, 355, 361, 365, 371, 377, 385

Offset: 1

Author

Jeff Burch

Keywords

• nonn
• nice

Comments

Or, composite numbers with smallest prime factor >= 5.

Or, nonprime numbers n such that binomial(n+3, 3) mod n == 1. - Hieronymus Fischer, Sep 30 2007

Note that the primes > 3 are congruent to +-1 mod 6.

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

Intersection of A002808 and A007310. - Reinhard Zumkeller, Jun 30 2012

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

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A171993 (nonprimes of the form 3*k+-1).

Cf. A000040, A133620-A133625, A133630, A133634-A133636.

Cf. A133873, A133883, A133880, A133890, A133900, A133910.

Cf. A069043, A067793 (composite n such that phi(n) > 2n/3).

Programs

Haskell

a038509 n = a038509_list !! (n-1)
a038509_list = [x | x <- a002808_list, gcd x 6 == 1]
-- Reinhard Zumkeller, Aug 05 2014, Jun 30 2012

Maple

A038509 := proc(n)
    option remember;
    if n = 1 then
        25;
    else
        for a from procname(n-1)+1 do
            if not isprime(a) and modp(a,6) in {1,5} then
                return a;
            end if;
        end do:
    end if;
end proc:
seq(A038509(n),n=1..30) ; # R. J. Mathar, Feb 28 2020

Mathematica

Select[Range[1000], FactorInteger[#][[1,1]] >= 5 && ! PrimeQ[#] &] (* Robert G. Wilson v, Dec 19 2009 *)
With[{nn=400},Select[Rest[Complement[Range[nn],Prime[Range[ PrimePi[ nn]]]]], MemberQ[ {1,5},Mod[#,6]]&]] (* Harvey P. Dale, Feb 21 2013 *)
Select[Range[400],CompositeQ[#]&&MemberQ[{1,5},Mod[#,6]]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, May 13 2019 *)

PARI

is(n)=gcd(n,6)==1 && !isprime(n) && n>7 \\ Charles R Greathouse IV, Nov 20 2012

Formula

a(n) ~ 3n. - Charles R Greathouse IV, Nov 20 2012

Extensions

More terms from Robert G. Wilson v, Dec 19 2009

Entry revised by N. J. A. Sloane, Dec 31 2011, at the suggestion of Gary Detlefs