A091684 -id:A091684 - cp's OEIS frontend

A046913 Sum of divisors of n not congruent to 0 mod 3.

1, 3, 1, 7, 6, 3, 8, 15, 1, 18, 12, 7, 14, 24, 6, 31, 18, 3, 20, 42, 8, 36, 24, 15, 31, 42, 1, 56, 30, 18, 32, 63, 12, 54, 48, 7, 38, 60, 14, 90, 42, 24, 44, 84, 6, 72, 48, 31, 57, 93, 18, 98, 54, 3, 72, 120, 20, 90, 60, 42, 62, 96, 8, 127, 84, 36, 68, 126

Offset: 1

N. J. A. Sloane

			Divisors of 12 are 1 2 3 4 6 12 and discarding 3 6 and 12 we get a(12) = 1 + 2 + 4 = 7.
x + 3*x^2 + x^3 + 7*x^4 + 6*x^5 + 3*x^6 + 8*x^7 + 15*x^8 + x^9 + 18*x^10 + ...

Seiichi Manyama, Table of n, a(n) for n = 1..10000
Hershel M. Farkas, On an arithmetical function, Ramanujan J., Vol. 8, No. 3 (2004), pp. 309-315.
Pavel Guerzhoy and Ka Lun Wong, Farkas' identities with quartic characters, The Ramanujan Journal (2020), preprint, arXiv:1905.06506 [math.NT], 2019.

Cf. A035191.

Cf. A000726, A051731, A091684, A000203, A002324.

[SumOfDivisors(3*k)-3*SumOfDivisors(k):k in [1..70]]; // Marius A. Burtea, Jun 01 2019

Table[DivisorSigma[1, 3*w]-3*DivisorSigma[1, w], {w, 1, 256}]
DivisorSum[#1, # &, Mod[#, 3] != 0 &] & /@ Range[68] (* Jayanta Basu, Jun 30 2013 *)
f[p_, e_] := If[p == 3, 1, (p^(e+1)-1)/(p-1)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 17 2020 *)

{a(n) = if( n<1, 0, sigma(3*n) - 3 * sigma(n))} /* Michael Somos, Jul 19 2004 */

a(n) = sigma(n \ 3^valuation(n, 3)) \\ David A. Corneth, Jun 01 2019

Multiplicative with a(3^e) = 1, a(p^e) = (p^(e+1)-1)/(p-1) for p<>3. - Vladeta Jovovic, Sep 11 2002
G.f.: Sum_{k>0} x^k*(1+2*x^k+2*x^(3*k)+x^(4*k))/(1-x^(3*k))^2. - Vladeta Jovovic, Dec 18 2002
a(n) = A000203(3n)-3*A000203(n). - Labos Elemer, Aug 14 2003
Inverse Mobius transform of A091684. - Gary W. Adamson, Jul 03 2008
Dirichlet g.f.: zeta(s)*zeta(s-1)*(1-1/3^(s-1)). - R. J. Mathar, Feb 10 2011
G.f. A(x) satisfies: 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w)= u^2 + 9 * v^2 + 16 * w^2 - 6 * u*v + 4 * u*w - 24 * v*w - v + w. - Michael Somos, Jul 19 2004
L.g.f.: log(Product_{k>=1} (1 - x^(3*k))/(1 - x^k)) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, Mar 14 2018
a(n) = A002324(n) + 3*Sum_{j=1, n-1} A002324(j)*A002324(n-j). See Farkas and Guerzhoy links. - Michel Marcus, Jun 01 2019
a(3*n) = a(n). - David A. Corneth, Jun 01 2019
Sum_{k=1..n} a(k) ~ Pi^2 * n^2 / 18. - Vaclav Kotesovec, Sep 17 2020

A068626 a(3n) = a(3n-1) = 3n^2, a(3n-2) = 3n^2 - 3*n + 1.

Original entry on oeis.org

1, 3, 3, 7, 12, 12, 19, 27, 27, 37, 48, 48, 61, 75, 75, 91, 108, 108, 127, 147, 147, 169, 192, 192, 217, 243, 243, 271, 300, 300, 331, 363, 363, 397, 432, 432, 469, 507, 507, 547, 588, 588, 631, 675, 675, 721, 768, 768, 817, 867, 867, 919, 972, 972, 1027, 1083, 1083, 1141

Offset: 1

Amarnath Murthy, Feb 26 2002

Or, a(1) = 1, a(n) = n + a(n-1) if n does not divide a(n-1) else a(n) = a(n-1). E.g. a(6) = a(5) = 12 as 6 divides 12. a(10) = 10+a(9) = 10+27 = 37.

B. D. Swan, Table of n, a(n) for n = 1..90000
Index entries for linear recurrences with constant coefficients, signature (1,0,2,-2,0,-1,1).

Cf. A091684 (first differences).

[(n mod 3 eq 1) select (n+2)^2/3 - n-1 else (n+((n mod 3)^2) mod 3 )^2/3  : n in [1..50]]; // Marius A. Burtea, Feb 19 2020

LinearRecurrence[{1,0,2,-2,0,-1,1},{1,3,3,7,12,12,19},60] (* Harvey P. Dale, Jun 29 2022 *)

Vec(x*(1 + 2*x + 2*x^3 + x^4) / ((1 - x)^3*(1 + x + x^2)^2) + O(x^50)) \\ Colin Barker, Feb 19 2020

my @a = (1); for (my $n = 1; $n <= 90000; $n ++) {
  $a[$n] = $a[$n - 1] + ($a[$n - 1] % $n != 0 ? $n : 0);
  print "$n $a[$n]\n";
} # Georg Fischer Feb 18 2020

From Colin Barker, Feb 18 2020: (Start)
G.f.: x*(1 + 2*x + 2*x^3 + x^4) / ((1 - x)^3*(1 + x + x^2)^2).
a(n) = a(n-1) + 2*a(n-3) - 2*a(n-4) - a(n-6) + a(n-7) for n>7.
(End)
Sum_{n>=1} 1/a(n) = Pi/sqrt(3)*tanh(Pi/(2*sqrt(3))) + Pi^2/9. - Amiram Eldar, Sep 21 2023

Entry revised by N. J. A. Sloane, Mar 13 2006

A100050 A Chebyshev transform of n.

Original entry on oeis.org

0, 1, 2, 0, -4, -5, 0, 7, 8, 0, -10, -11, 0, 13, 14, 0, -16, -17, 0, 19, 20, 0, -22, -23, 0, 25, 26, 0, -28, -29, 0, 31, 32, 0, -34, -35, 0, 37, 38, 0, -40, -41, 0, 43, 44, 0, -46, -47, 0, 49, 50, 0, -52, -53, 0, 55, 56, 0, -58, -59, 0, 61, 62, 0, -64, -65, 0, 67, 68, 0, -70, -71, 0, 73, 74, 0, -76, -77, 0, 79, 80, 0, -82, -83, 0, 85, 86, 0

Offset: 0

Paul Barry, Oct 31 2004

A Chebyshev transform of x/(1-x)^2: if A(x) is the g.f. of a sequence, map it to ((1-x^2)/(1+x^2))A(x/(1+x^2)).

			x + 2*x^2 - 4*x^4 - 5*x^5 + 7*x^7 + 8*x^8 - 10*x^10 - 11*x^11 + 13*x^13 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Michael Somos, Rational Function Multiplicative Coefficients
Index entries for linear recurrences with constant coefficients, signature (2,-3,2,-1).

Cf. A165202 (partial sums).

Cf. A099837, A099443, A011655, A100047, A100048, A100051, A091684.

LinearRecurrence[{2, -3, 2, -1}, {0, 1, 2, 0},50] (* G. C. Greubel, Aug 08 2017 *)

{a(n) = n * (-1)^(n\3) * sign( n%3)} /* Michael Somos, Mar 19 2011 */

{a(n) = local(A, p, e); if( abs(n)<1, 0, A = factor(abs(n)); prod( k=1, matsize(A)[1], if( p=A[k,1], e=A[k,2]; if( p==2, -(-2)^e, (kronecker( -12, p) * p)^e))))} /* Michael Somos, Mar 19 2011 */

[lucas_number1(n,2,1)*lucas_number1(n,1,1) for n in range(0,88)] # Zerinvary Lajos, Jul 06 2008

Euler transform of length 6 sequence [ 2, -3, -2, 0, 0, 2]. - Michael Somos, Mar 19 2011
a(n) is multiplicative with a(2^e) = -(-2)^e if e>0, a(3^e) = 0^e, a(p^e) = p^e if p == 1 (mod 6), a(p^e) = (-p)^e if p == 5 (mod 6). - Michael Somos, Mar 19 2011
G.f.: x*(1 - x^2)^3 *(1 - x^3)^2 / ((1 - x)^2 *(1 - x^6)^2) = x *(1 + x)^2 *(1 - x^2) / (1 + x^3)^2. - Michael Somos, Mar 19 2011
a(3*n) = 0, a(3*n + 1) = (-1)^n * (3*n + 1), a(3*n + 2) = (-1)^n * (3*n + 2). a(-n) = a(n). - Michael Somos, Mar 19 2011
G.f.: x(1-x^2)/(1-x+x^2)^2.
a(n) = 2*a(n-1) -3*a(n-2) +2*a(n-3) -a(n-4).
a(n) = n*Sum_{k=0..floor(n/2)} (-1)^k*binomial(n-k,k)*(n-2k)/(n-k).

A091703 Count, setting 5n to zero.

Original entry on oeis.org

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

Paul Barry, Jan 30 2004

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).

Cf. A001477, A011558, A080891, A091684, A011558.

[(n mod 5 eq 0) select 0 else n: n in [0..80]]; // G. C. Greubel, Feb 28 2019

Table[If[Divisible[n,5],0,n],{n,0,80}] (* Harvey P. Dale, Apr 26 2018 *)

a(n) = if (n % 5, n, 0); \\ Michel Marcus, Feb 28 2019

[n if not n % 5==0 else 0 for n in range(80)] # G. C. Greubel, Feb 28 2019

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.
a(n) = |A080891(n)|*A001477(n). (End)
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

A162397 a(n) = n * Kronecker(-3, n).

Original entry on oeis.org

1, -2, 0, 4, -5, 0, 7, -8, 0, 10, -11, 0, 13, -14, 0, 16, -17, 0, 19, -20, 0, 22, -23, 0, 25, -26, 0, 28, -29, 0, 31, -32, 0, 34, -35, 0, 37, -38, 0, 40, -41, 0, 43, -44, 0, 46, -47, 0, 49, -50, 0, 52, -53, 0, 55, -56, 0, 58, -59, 0, 61, -62, 0, 64, -65, 0

Offset: 1

Michael Somos, Jul 02 2009

In Gil and Robins 2003 on page 33 the g.f. is denoted by f_{4, 2}(x). - Michael Somos, Sep 04 2015

			G.f. = x - 2*x^2 + 4*x^4 - 5*x^5 + 7*x^7 - 8*x^8 + 10*x^10 - 11*x^11 + ...

George E. Andrews and Bruce C. Berndt, Ramanujan's lost notebook, Part I, Springer, New York, 2005, MR2135178 (2005m:11001) See p. 319, Equation (14.3.6).

Juan B. Gil and Sinai Robins, Hecke operators on rational functions, arXiv:math/0309244 [math.NT], 2003.

Cf. A016777, A016789, A091684, A102283.

Table[n*KroneckerSymbol[-3,n],{n,80}] (* Harvey P. Dale, Mar 14 2015 *)

PARI
```
{a(n) = n * kronecker(-3, n)};
```

Euler transform of length 3 sequence [ -2, -1, 2].
a(n) is completely multiplicative with a(3^e) = 0^e, a(p^e) = p^e if p == 1 (mod 3), a(p^e) = (-p)^e if p == 2 (mod 3).
G.f.: (x - x^3) / (1 + x + x^2)^2.
a(3*n) = 0. a(n) = a(-n). abs(a(n)) = A091684(n). a(n) = n * A102283(n). a(3*n + 1) = A016777(n). a(3*n + 2) = - A016789(n). - Michael Somos, Mar 14 2012
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2/3. - Amiram Eldar, Nov 23 2023

cp's OEIS Frontend

A046913 Sum of divisors of n not congruent to 0 mod 3.

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A068626 a(3n) = a(3n-1) = 3*n^2, a(3n-2) = 3*n^2 - 3*n + 1.

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A100050 A Chebyshev transform of n.

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A091703 Count, setting 5n to zero.

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A162397 a(n) = n * Kronecker(-3, n).

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A068626 a(3n) = a(3n-1) = 3n^2, a(3n-2) = 3n^2 - 3*n + 1.