Search: a035186 -id:a035186
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A035184
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a(n) = Sum_{d|n} Kronecker(-1, d).
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+10
6
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1, 2, 0, 3, 2, 0, 0, 4, 1, 4, 0, 0, 2, 0, 0, 5, 2, 2, 0, 6, 0, 0, 0, 0, 3, 4, 0, 0, 2, 0, 0, 6, 0, 4, 0, 3, 2, 0, 0, 8, 2, 0, 0, 0, 2, 0, 0, 0, 1, 6, 0, 6, 2, 0, 0, 0, 0, 4, 0, 0, 2, 0, 0, 7, 4, 0, 0, 6, 0, 0, 0, 4, 2, 4, 0, 0, 0, 0, 0, 10, 1, 4, 0, 0, 4, 0, 0, 0, 2, 4, 0, 0, 0, 0, 0, 0, 2, 2, 0, 9, 2, 0, 0, 8, 0
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OFFSET
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1,2
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LINKS
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FORMULA
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a(n) is multiplicative with a(2^e) = e + 1, a(p^e) = e + 1 if p == 1 (mod 4), a(p^e) = (1 + (-1)^e) / 2 if p == 3 (mod 4). - Michael Somos, Jan 05 2012
a(4*n + 3) = a(9*n + 3) = a(9*n + 6) = 0. a(4*n + 1) = A008441(n). a(8*n + 1) = A113407(n). a(8*n + 5) = 2 * A053692(n). a(12*n + 1) = A002175(n). a(12*n + 5) = 2 * A121444(n).
Dirichlet g.f.: zeta(s)*beta(s)/(1 - 2^(-s)), where beta is the Dirichlet beta function. - Ralf Stephan, Mar 27 2015
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/2 = 1.570796... (A019669). - Amiram Eldar, Oct 17 2022
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EXAMPLE
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G.f. = x + 2*x^2 + 3*x^4 + 2*x^5 + 4*x^8 + x^9 + 4*x^10 + 2*x^13 + 5*x^16 + 2*x^17 + ...
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MATHEMATICA
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PROG
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(PARI) {a(n) = if( n<1, 0, direuler( p=2, n, 1/((1 - X) * (1 - kronecker( -1, p) * X))) [n])}; /* Michael Somos, Jan 05 2012 */
(PARI) {a(n) = if( n<1, 0, sumdiv( n, d, kronecker( -1, d)))}; /* Michael Somos, Jan 05 2012 */
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CROSSREFS
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Inverse Moebius transform of A034947.
Sum_{d|n} Kronecker(k, d): A035143..A035181 (k=-47..-9, skipping numbers that are not cubefree), A035182 (k=-7), A192013 (k=-6), A035183 (k=-5), A002654 (k=-4), A002324 (k=-3), A002325 (k=-2), this sequence (k=-1), A000012 (k=0), A000005 (k=1), A035185 (k=2), A035186 (k=3), A001227 (k=4), A035187..A035229 (k=5..47, skipping numbers that are not cubefree).
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KEYWORD
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nonn,easy,mult
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AUTHOR
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STATUS
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approved
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A035181
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a(n) = Sum_{d|n} Kronecker(-9, d).
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+10
4
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1, 2, 1, 3, 2, 2, 0, 4, 1, 4, 0, 3, 2, 0, 2, 5, 2, 2, 0, 6, 0, 0, 0, 4, 3, 4, 1, 0, 2, 4, 0, 6, 0, 4, 0, 3, 2, 0, 2, 8, 2, 0, 0, 0, 2, 0, 0, 5, 1, 6, 2, 6, 2, 2, 0, 0, 0, 4, 0, 6, 2, 0, 0, 7, 4, 0, 0, 6, 0, 0, 0, 4, 2, 4, 3, 0, 0, 4, 0, 10, 1, 4, 0, 0, 4, 0, 2, 0, 2, 4, 0, 0, 0, 0, 0, 6, 2, 2, 0, 9, 2, 4, 0, 8, 0
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OFFSET
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1,2
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LINKS
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FORMULA
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a(n) is multiplicative with a(2^e) = e + 1, a(3^e) = 1, a(p^e) = e + 1 if p == 1 (mod 4), a(p^e) = (1 + (-1)^e) / 2 if p == 3 (mod 4) and p > 3.
Dirichlet g.f.: zeta(s) * L(chi,s) where chi(n) = Kronecker(-9, n). Sum_{n>0} a(n) / n^s = Product_{p prime} 1 / ((1 - p^-s) * (1 - Kronecker(-9, p) * p^-s)). (End)
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2*Pi/3 = 2.094395... (A019693). - Amiram Eldar, Oct 17 2022
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EXAMPLE
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x + 2*x^2 + x^3 + 3*x^4 + 2*x^5 + 2*x^6 + 4*x^8 + x^9 + 4*x^10 + 3*x^12 + ...
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MATHEMATICA
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a[ n_] := If[ n < 1, 0, Sum[ KroneckerSymbol[ -9, d], { d, Divisors[ n]}]] (* Michael Somos, Jun 24 2011 *)
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PROG
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(PARI) {a(n) = if( n<1, 0, sumdiv( n, d, kronecker( -9, d)))} \\ Michael Somos, Jun 24 2011
(PARI) {a(n) = if( n<1, 0, direuler( p=2, n, 1 / ((1 - X) * (1 - kronecker( -9, p) * X))) [n])} \\ Michael Somos, Jun 24 2011
(PARI) {a(n) = local(A, p, e); if( n<0, 0, A = factor(n); prod(k=1, matsize(A)[1], if(p = A[k, 1], e = A[k, 2]; if( p==2, e+1, if( p==3, 1, if( p%4==1, e+1, (1 + (-1)^e)/2))))))} \\ Michael Somos, Jun 24 2011
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CROSSREFS
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Sum_{d|n} Kronecker(k, d): A035143..A035181 (k=-47..-9, skipping numbers that are not cubefree), A035182 (k=-7), A192013 (k=-6), A035183 (k=-5), A002654 (k=-4), A002324 (k=-3), A002325 (k=-2), A035184 (k=-1), A000012 (k=0), A000005 (k=1), A035185 (k=2), A035186 (k=3), A001227 (k=4), A035187..A035229 (k=5..47, skipping numbers that are not cubefree).
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KEYWORD
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nonn,mult
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AUTHOR
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STATUS
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approved
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A327785
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Square array read by antidiagonals: A(n,k) = Sum_{d|n} (k/d), (n>=1, k>=0), where (m/n) is the Kronecker symbol.
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+10
1
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1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 0, 0, 3, 1, 1, 1, 1, 1, 2, 1, 1, 0, 2, 1, 0, 4, 1, 1, 1, 0, 1, 0, 0, 2, 1, 1, 2, 1, 1, 2, 0, 2, 4, 1, 1, 1, 2, 1, 1, 2, 0, 1, 3, 1, 1, 2, 0, 3, 2, 0, 2, 0, 1, 4, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 0, 2, 1, 1, 0, 2, 3, 0, 4, 0, 0, 3, 0, 0, 6, 1
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OFFSET
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1,5
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LINKS
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EXAMPLE
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Square array begins:
1, 1, 1, 1, 1, 1, 1, 1, ...
1, 2, 1, 0, 1, 0, 1, 2, ...
1, 2, 0, 1, 2, 0, 1, 2, ...
1, 3, 1, 1, 1, 1, 1, 3, ...
1, 2, 0, 0, 2, 1, 2, 0, ...
1, 4, 0, 0, 2, 0, 1, 4, ...
1, 2, 2, 0, 2, 0, 0, 1, ...
1, 4, 1, 0, 1, 0, 1, 4, ...
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MATHEMATICA
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A[n_, k_] := Sum[KroneckerSymbol[k, d], {d, Divisors[n]}];
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CROSSREFS
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Columns k=0..31 give A000012, A000005, A035185, A035186, A001227, A035187, A035188, A035189, A035185, A035191, A035192, A035193, A035194, A035195, A035196, A035197, A001227, A035199, A035200, A035201, A035202, A035203, A035204, A035205, A035188, A035207, A035208, A035186, A035210, A035211, A035212, A035213.
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KEYWORD
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AUTHOR
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STATUS
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approved
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A035252
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Indices of the nonzero terms in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)*p^(-s)+Kronecker(m,p)*p^(-2s))^(-1) for m= 3.
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+10
0
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1, 3, 4, 9, 11, 12, 13, 16, 23, 25, 27, 33, 36, 37, 39, 44, 47, 48, 49, 52, 59, 61, 64, 69, 71, 73, 75, 81, 83, 92, 97, 99, 100, 107, 108, 109, 111, 117, 121, 131, 132, 141, 143, 144, 147, 148, 156, 157, 167, 169, 176, 177, 179, 181, 183, 188, 191, 192, 193
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graph;
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OFFSET
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1,2
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LINKS
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PROG
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(PARI) m=3; select(x -> x, direuler(p=2, 101, 1/(1-(kronecker(m, p)*(X-X^2))-X)), 1)
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CROSSREFS
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Cf. A035186 (the expansion itself).
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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