A091703 Count, setting 5n to zero.
0, 1, 2, 3, 4, 0, 6, 7, 8, 9, 0, 11, 12, 13, 14, 0, 16, 17, 18, 19, 0, 21, 22, 23, 24, 0, 26, 27, 28, 29, 0, 31, 32, 33, 34, 0, 36, 37, 38, 39, 0, 41, 42, 43, 44, 0, 46, 47, 48, 49, 0, 51, 52, 53, 54, 0, 56, 57, 58, 59, 0, 61, 62, 63, 64, 0, 66, 67, 68, 69, 0, 71, 72, 73, 74, 0, 76, 77
Offset: 0
Links
- G. C. Greubel, Table of n, a(n) for n = 0..10000
- Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,2,0,0,0,0,-1).
Programs
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Magma
[(n mod 5 eq 0) select 0 else n: n in [0..80]]; // G. C. Greubel, Feb 28 2019
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Mathematica
Table[If[Divisible[n,5],0,n],{n,0,80}] (* Harvey P. Dale, Apr 26 2018 *) -
PARI
a(n) = if (n % 5, n, 0); \\ Michel Marcus, Feb 28 2019
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Sage
[n if not n % 5==0 else 0 for n in range(80)] # G. C. Greubel, Feb 28 2019
Formula
a(n) = Product_{k=0..4} Sum_{j=1..n} e^(2*Pi*ijk/5), i=sqrt(-1).
a(n) = cos(4*Pi*n/5 + 2*Pi/5)*(n*cos(2*Pi*n/5 + Pi/5)/5 + n*sqrt(1/5 - 2*sqrt(5)/25)*sin(2*Pi*n/5 + Pi/5) + n*(1/5 - sqrt(5)/5)) + sin(4*Pi*n/5 + 2*Pi/5)*(n*sqrt(2*sqrt(5)/25 + 1/5)*cos(2*Pi*n/5 + Pi/5) + sqrt(5)*n*sin(2*Pi*n/5 + Pi/5)/5 - n*sqrt(2*sqrt(5)/25 + 2/5)) - n*(sqrt(5)/5 + 1/5)*cos(2*Pi*n/5 + Pi/5) - n*sqrt(2/5 - 2*sqrt(5)/25)*sin(2*Pi*n/5 + Pi/5) + 4*n/5.
a(n) = n^5 mod (5*n). - Paul Barry, Apr 13 2005
Multiplicative with a(5^e) = 0, a(p^e) = p^e otherwise. - Mitch Harris, Jun 09 2005
From R. J. Mathar, Feb 04 2009: (Start)
a(n) = 2*a(n-5) - a(n-10).
G.f.: x*(1 + 2*x + 3*x^2 + 4*x^3 + 4*x^5 + 3*x^6 + 2*x^7 + x^8)/(1-x^5)^2.
From R. J. Mathar, Apr 14 2011: (Start)
a(n) = n*A011558(n).
Dirichlet g.f.: (1-5^(1-s))*zeta(s-1). (End)
Sum_{k=1..n} a(k) ~ (2/5) * n^2. - Amiram Eldar, Nov 20 2022