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A007310
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Numbers congruent to 1 or 5 mod 6.
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87
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1, 5, 7, 11, 13, 17, 19, 23, 25, 29, 31, 35, 37, 41, 43, 47, 49, 53, 55, 59, 61, 65, 67, 71, 73, 77, 79, 83, 85, 89, 91, 95, 97, 101, 103, 107, 109, 113, 115, 119, 121, 125, 127, 131, 133, 137, 139, 143, 145, 149
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| 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]
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]
Numbers n such that ChebyshevT[x, x/2] is not an integer (is integer/2). [From Artur Jasinski (grafix(AT)csl.pl), Feb 13 2010]
If 12k+1 is a pefect square..(0,2,4,10,14,24,30,44...) then the square root of 12k+1 = a(n) [From Gary Detlefs (gdetlefs(AT)aol.com), Feb 22 2010]
A089128(a(n)) = 1. Complement of A047229(n+1) for n>=1. See A164576 for corresponding values A175485(a(n)). [From Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), May 28 2010]
Cf. property described by G. Detlefs in A113801 and in Comment: more generally, these numbers are of the form (2*h*n+(h-4)*(-1)^n-h)/4 (h, n natural numbers), then ((2*h*n+(h-4)*(-1)^n-h)/4)^2-1==0 (mod h); in this case, a(n)^2-1==0 (mod 6). Also a(n)^2-1==0 (mod 12). [From Bruno Berselli (berselli.bruno(AT)yahoo.it), Nov 05 2010 - Nov 17 2010]
Numbers n such that sum(k^14,k=1..n) mod n = 0. (Conjectured) [From Gary Detlefs, Dec 27, 2011]
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REFERENCES
| L. B. W. Jolley, "Summation of Series", Dover Publications, 1961.
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LINKS
| Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
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
Eric Weisstein, Pi Formulas [From Jaume Oliver Lafont (joliverlafont(AT)gmail.com), Oct 23 2009]
Index entries for sequences related to smooth numbers [From Michael Porter (michael_b_porter(AT)yahoo.com), Oct 09 2009]
Index entries for sequences related to linear recurrences with constant coefficients, signature (1,1,-1).
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FORMULA
| a(n) = (6n + (-1)^n - 3)/2. - 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(n) = a(n-1) + a(n-2) - a(n-3), n>=4. - 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
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). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Oct 02 2008
1 + 1/5 - 1/7 - 1/11 + + - - ... = Pi/3 = A019670 [Jolley eq (315)]. - Jaume Oliver Lafont (joliverlafont(AT)gmail.com), Oct 23 2009
A169611(a(n)) = 0. - Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Dec 03 2009
a(n) = ( 6*A062717(n)+1 )^(1/2). - Gary Detlefs (gdetlefs(AT)aol.com), Feb 22 2010
a(n) = 6*A000217(n-1)+1 - 2*sum(a(i), i=1..n-1) (n>1). [From Bruno Berselli (berselli.bruno(AT)yahoo.it), Nov 05 2010]
a(n) = 6*n-a(n-1)-6 (with a(1)=1). - Vincenzo Librandi, Nov 18 2010
Sum_{n>=1} (-1)^(n+1)/a(n) = A093766 [Jolley eq (84)]. - R. J. Mathar, Mar 24 2011
a(n) = 6*floor(n/2)+(-1)^(n+1). - Gary Detlefs, Dec 29 2011
a(n) = 3*n + ((n+1) mod 2) - 2. - Gary Detlefs, Jan 08 2012
a(n) = 2*n+1 + 2*floor((n-2)/2) = 2*n-1 + 2*floor(n/2), leading to the o.g.f. given by R. J. Mathar above. - Wolfdieter Lang, Jan 20 2012
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MAPLE
| P:=(j, n)-> sum(k^j, k=1..n): for n from 1 to 149 do if (P(14, n) mod n = 0) then print(n) fi od; [From Gary Detlefs, Dec 27 2011]
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MATHEMATICA
| aa = {}; Do[If[IntegerQ[ChebyshevT[x, x/2]], , AppendTo[aa, x]], {x, 0, 150}]; aa (*Artur Jasinski*) [From Artur Jasinski (grafix(AT)csl.pl), Feb 13 2010]
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PROG
| (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]
(Haskell)
a007310 n = a007310_list !! (n-1)
a007310_list = 1 : 5 : map (+ 6) a007310_list
-- Reinhard Zumkeller, Jan 07 2012
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CROSSREFS
| Union of A016921 and A016969. Essentially the same as A038179.
Cf. A000330, A144065, A047209, A047336.
For k-rough numbers with other values of k, see A000027, A005408, A007310, A007775, A008364-A008366, A166061, A166063.
Subsequence of A186422.
Cf. A032528 (partial sums).
Cf. A047522, A056020, A090771, A091998, A175885-A175887.
Sequence in context: A136801 A106571 A067291 * A069040 A070191 A135775
Adjacent sequences: A007307 A007308 A007309 * A007311 A007312 A007313
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KEYWORD
| nonn,easy
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AUTHOR
| C. Christofferson (Magpie56(AT)aol.com)
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