1,2

Or, numbers relatively prime to 2 and 3.

Apart from initial term(s), dimension of the space of weight 2n cuspidal newforms for Gamma_0( 38 ).

Numbers k such that k mod 2 = 1 and (k+1) mod 3 <> 1. - Klaus Brockhaus (klaus-brockhaus(AT)t-online.de), Jun 15 2004

Also numbers n such that the sum of the squares of the first n integers is divisible by n, or A000330(n) = n(n+1)(2n+1)/6 is divisible by n. - Alexander Adamchuk (alex(AT)kolmogorov.com), Jan 04 2007

Or, except for the first term, numbers the least prime factor of which is >=5. - Zak Seidov (zakseidov(AT)yahoo.com), Apr 26 2007

A126759(a(n)) = n+1. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Jun 16 2008

Terms of this sequence (starting from the second term) are equal to the result of the expression sqrt(4!*(k+1) + 1) - but only when this expression yields integral values (that is when the parameter k takes values, which are terms of A144065) [From Alexander R. Povolotsky (pevnev(AT)juno.com), Sep 09 2008]

For n>1: a(n) is prime iff A075743(n-2) = 1; a(2*n-1)=A016969(n-1), a(2*n)=A016921(n-1). [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Oct 02 2008]

Also numbers n such that [sum_i=1..n {i^2}] / n = c, c an integer. [From Ctibor O. Zizka (c.zizka(AT)email.cz), Oct 03 2008]

A156543 is a subsequence. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Feb 10 2009]

Also the 5-rough numbers: positive integers that have no prime factors less than 5 [From Michael Porter (michael_b_porter(AT)yahoo.com), Oct 09 2009]

Apart from initial term(s), a(n)=6*n-a(n-1), (with a(1)=5). Conjecture: Apart from initial couple (3-5), in this sequence are all couple of prime in the form p+2. [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Oct 18 2009]

n withaut trivial prime divisors; also, (number of trivial prime divisors of n)<1. [From Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Dec 03 2009]

L. B. W. Jolley, "Summation of Series", Dover Publications, 1961, p. 62.

Index entries for sequences related to linear recurrences with constant coefficients

William A. Stein, Dimensions of the spaces S_k^{new}(Gamma_0(N))

William A. Stein, The modular forms database

Eric Weisstein's World of Mathematics, Rough Number

Index entries for sequences related to smooth numbers [From Michael Porter (michael_b_porter(AT)yahoo.com), Oct 09 2009]

Eric Weisstein, Pi Formulas [From Jaume Oliver Lafont (joliverlafont(AT)gmail.com), Oct 23 2009]

a(n) = (6n +(-1)^n - 3)/2; n > 0. - Antonio Esposito (antonio.b.esposito(AT)italtel.it), Jan 18 2002

n such that phi(4n)=phi(3n) - Benoit Cloitre (benoit7848c(AT)orange.fr), Aug 06 2003

a(1)=1, a(2)=5, a(3)=7, a(n) = a(n-1) + a(n-2) - a(n-3) - Roger Bagula (rlbagulatftn(AT)yahoo.com)

a(n)=3n-1-(n mod 2), n=1, 2, ... - Zak Seidov, Jan 18 2006

a(1)=1 then alternatively add 4 and 2. a(1)=1, a(n)=a(n-1)+3+(-1)^n. - Zak Seidov (zakseidov(AT)yahoo.com), Mar 25 2006

1 + 1/5^2 + 1/7^2 + 1/11^2 + ...= (Pi)^2/9 [Jolley] - Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 20 2006

a(1)=1, a(2)=5; for n>=3 a(n)=a(n-2)+6 - Zak Seidov (zakseidov(AT)gmail.com), Apr 18 2007

Expand (x+x^5)/(1-x^6) = x +x^5 +x^7 +x^11 +x^13+... O.g.f.: x(1+4x+x^2)/((1+x)(1-x)^2). - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), May 23 2008

a(n) = 6*floor(n/2) - 1 + 2*(n mod 2). [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Oct 02 2008]

1 + 1/5 - 1/7 - 1/11 + + - - ... = Pi/3 [From Jaume Oliver Lafont (joliverlafont(AT)gmail.com), Oct 23 2009]

a(n)=6*n-a(n-1)-6 (with a(1)=1) [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Nov 23 2009]

A169611(a(n))=0. [From Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Dec 03 2009]

For n=2, a(2)=6*2-1-6=5; n=3, a(3)=6*3-5-6=7; n=4, a(4)=6*4-7-6=11 [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Nov 23 2009]

(PARI) The following PARI program applies to generate all terms besides first one: j=[]; for(n=0, 1000, if((floor(sqrt(4!*(n+1) + 1))) == ceil(sqrt(4!*(n+1) + 1)), j=concat(j, floor(sqrt(4!*(n+1) + 1))))); j [From Alexander R. Povolotsky (pevnev(AT)juno.com), Sep 09 2008]

(Other) sage: [i for i in range(150) if gcd(6, i) == 1] [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 21 2009]

(PARI) isA007310(n) = gcd(n, 6)==1 [From Michael Porter (michael_b_porter(AT)yahoo.com), Oct 09 2009]

Cf. A000330. A038179 is the same, apart from the first two terms.

A144065 [From Alexander R. Povolotsky (pevnev(AT)juno.com), Sep 09 2008]

Union of A016921 and A016969. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Oct 02 2008]

For k-rough numbers with other values of k, see A000027 A005408 A007310 A007775 A008364 A008365 A008366 A166061 A166063 [From Michael Porter (michael_b_porter(AT)yahoo.com), Oct 09 2009]

Sequence in context: A136801 A106571 A067291 this_sequence A069040 A070191 A135775

Adjacent sequences: A007307 A007308 A007309 this_sequence A007311 A007312 A007313

Cf. If A168546=the nontrivial primes, then 2 and 3 are trivial primes. A169611. [From Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Dec 03 2009]

nonn,easy,new

C. Christofferson (Magpie56(AT)aol.com)