A078182 -id:A078182 - cp's OEIS frontend

A078181 a(n) = Sum_{d|n, d == 1 (mod 3)} d.

1, 1, 1, 5, 1, 1, 8, 5, 1, 11, 1, 5, 14, 8, 1, 21, 1, 1, 20, 15, 8, 23, 1, 5, 26, 14, 1, 40, 1, 11, 32, 21, 1, 35, 8, 5, 38, 20, 14, 55, 1, 8, 44, 27, 1, 47, 1, 21, 57, 36, 1, 70, 1, 1, 56, 40, 20, 59, 1, 15, 62, 32, 8, 85, 14, 23, 68, 39, 1, 88, 1, 5, 74, 38, 26, 100, 8, 14, 80, 71, 1

Offset: 1

Vladeta Jovovic, Nov 21 2002

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A001817, A001822, A035382, A051731, A272716, A353908.

Cf. Sum_{d|n, d==1 mod k} d: A000593 (k=2), this sequence (k=3), A050449 (k=4), A284097 (k=5), A284098 (k=6), A284099 (k=7), A284100 (k=8).

A078181 := proc(n)
    a := 0 ;
    for d in numtheory[divisors](n) do
        if modp(d,3) =1 then
            a :=a+d ;
        end if;
    end do:
    a;
end proc: # R. J. Mathar, May 11 2016

a[n_] := Plus @@ Select[Divisors[n], Mod[#, 3] == 1 &]; Array[a, 100] (* Giovanni Resta, May 11 2016 *)

G.f.: Sum_{n>=0} (3*n+1)*x^(3*n+1)/(1-x^(3*n+1)).
G.f.: -q*P'/P where P = Product_{n>=0} (1 - q^(3*n+1)). - Joerg Arndt, Aug 03 2011
Conjecture. If a(n)=n+1 then n==1 (mod 3). (Is this easy to settle? It has been verified for n=1,2,3,...,2000.) - John W. Layman, Apr 03 2006
The conjecture is false. The first and only counterexample below 10^8 is a(6800) = 6801 and 6800 == 2 (mod 3). - Lambert Herrgesell (zero815(AT)googlemail.com), May 06 2008
Equals A051731 * [1, 0, 0, 4, 0, 0, 7, 0, 0, 10, ...]. - Gary W. Adamson, Nov 06 2007
A272027(n/3) + a(n) + A078182(n) = A000203(n). - R. J. Mathar, May 25 2020
G.f.: Sum_{n >= 1} x^n*(1 + 2*x^(3*n))/(1 - x^(3*n))^2. - Peter Bala, Dec 19 2021
Sum_{k=1..n} a(k) = c * n^2 + O(n*log(n)), where c = Pi^2/36 = 0.274155... (A353908). - Amiram Eldar, Nov 26 2023

A050452 a(n) = Sum_{d|n, d == 3 (mod 4)} d.

Original entry on oeis.org

0, 0, 3, 0, 0, 3, 7, 0, 3, 0, 11, 3, 0, 7, 18, 0, 0, 3, 19, 0, 10, 11, 23, 3, 0, 0, 30, 7, 0, 18, 31, 0, 14, 0, 42, 3, 0, 19, 42, 0, 0, 10, 43, 11, 18, 23, 47, 3, 7, 0, 54, 0, 0, 30, 66, 7, 22, 0, 59, 18, 0, 31, 73, 0, 0, 14, 67, 0, 26, 42, 71, 3, 0, 0, 93, 19, 18

Offset: 1

N. J. A. Sloane, Dec 23 1999

Seiichi Manyama, Table of n, a(n) for n = 1..10000 (first 1000 terms from Harvey P. Dale)
Mariusz Skałba, A Note on Sums of Two Squares and Sum-of-divisors Functions, INTEGERS 20A (2020) A92.

Cf. A000593, A050449, A001842, A035462, A245058.

Cf. Sum_{d|n, d=k-1 mod k} d: A000593 (k=2), A078182 (k=3), this sequence (k=4).

A050452 := proc(n)
        a := 0 ;
        for d in numtheory[divisors](n) do
                if d mod 4 = 3 then
                        a := a+d ;
                end if;
        end do:
        a;
end proc:
seq(A050452(n),n=1..40) ; # R. J. Mathar, Dec 20 2011

Table[Total[Select[Divisors[n],Mod[#,4]==3&]],{n,80}] (* Harvey P. Dale, Jul 07 2013 *)

a(n) = sumdiv(n, d, d*((d % 4) == 3)); \\ Amiram Eldar, Nov 26 2023

a(n) = A000593(n) - A050449(n). - Reinhard Zumkeller, Apr 18 2006
G.f.: Sum_{k>=1} (4*k - 1)*x^(4*k-1)/(1 - x^(4*k-1)). - Ilya Gutkovskiy, Mar 21 2017
Sum_{k=1..n} a(k) = c * n^2 + O(n*log(n)), where c = Pi^2/48 = 0.205616... (A245058). - Amiram Eldar, Nov 26 2023

A035386 Number of partitions of n into parts congruent to 2 mod 3.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 1, 1, 2, 1, 3, 2, 3, 3, 4, 4, 6, 5, 7, 7, 9, 9, 12, 11, 15, 15, 18, 19, 23, 23, 29, 29, 35, 37, 43, 45, 53, 55, 64, 68, 78, 82, 95, 99, 114, 121, 136, 145, 164, 173, 196, 208, 232, 248, 276, 294, 328, 349, 386, 413, 456, 486, 537, 572, 629, 673, 737, 787

Offset: 0

Olivier Gérard

a(n) = A116376(3*n). - Reinhard Zumkeller, Feb 15 2006

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
James Mc Laughlin, Andrew V. Sills, Peter Zimmer, Rogers-Ramanujan-Slater Type Identities , arXiv:1901.00946 [math.NT]

Cf. A035382, A035451, A262928.

g:=add(x^(n*(3*n-1))/mul((1-x^(3*k))*(1-x^(3*k-1)), k = 1..n), n = 0..6): gser:=series(g,x,101): seq(coeff(gser,x,n), n = 0..100); # Peter Bala, Feb 02 2021

nmax=100; CoefficientList[Series[Product[1/(1-x^(3*k+2)),{k, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Feb 26 2015 *)
nmax = 100; poly = ConstantArray[0, nmax + 1]; poly[[1]] = 1; poly[[2]] = 0; Do[If[Mod[k, 3] == 2, Do[poly[[j + 1]] -= poly[[j - k + 1]], {j, nmax, k, -1}];], {k, 2, nmax}]; poly2 = Take[poly, {2, nmax + 1}]; poly3 = 1 + Sum[poly2[[n]]*x^n, {n, 1, Length[poly2]}]; CoefficientList[Series[1/poly3, {x, 0, Length[poly2]}], x] (* Vaclav Kotesovec, Jan 13 2017 *)
nmax = 50; s = Range[0, nmax/3]*3 + 2;
Table[Count[IntegerPartitions@n, x_ /; SubsetQ[s, x]], {n, 0, nmax}] (* Robert Price, Aug 05 2020 *)

{a(n)= if( n<0, 0, polcoeff( 1 / prod( k=1, n, 1 - (k%3==2) * x^k, 1 + x * O(x^n)), n))} /* Michael Somos, Jul 24 2007 */

a(n) = 1/n*Sum_{k=1..n} A078182(k)*a(n-k), a(0) = 1. - Vladeta Jovovic, Nov 21 2002
Euler transform of period 3 sequence [ 0, 1, 0, ...]. - Michael Somos, Jul 24 2007
a(n) ~ Gamma(2/3) * exp(sqrt(2*n)*Pi/3) / (2^(11/6) * sqrt(3) * Pi^(1/3) * n^(5/6)) * (1 + (Pi/72 - 5/(3*Pi)) / sqrt(2*n)). - Vaclav Kotesovec, Feb 26 2015, extended Jan 24 2017
G.f.: A(x) = Sum_{n >= 0} x^(n*(3*n-1))/Product_{k = 1..n} ((1 - x^(3*k)) *(1 - x^(3*k-1))). (Set z = x^2 and q = x^3 in Mc Laughlin et al., Section 1.3, Entry 7.) - Peter Bala, Feb 02 2021

A284103 a(n) = Sum_{d|n, d == 4 (mod 5)} d.

Original entry on oeis.org

0, 0, 0, 4, 0, 0, 0, 4, 9, 0, 0, 4, 0, 14, 0, 4, 0, 9, 19, 4, 0, 0, 0, 28, 0, 0, 9, 18, 29, 0, 0, 4, 0, 34, 0, 13, 0, 19, 39, 4, 0, 14, 0, 48, 9, 0, 0, 28, 49, 0, 0, 4, 0, 63, 0, 18, 19, 29, 59, 4, 0, 0, 9, 68, 0, 0, 0, 38, 69, 14, 0, 37, 0, 74, 0, 23, 0, 39, 79

Offset: 1

Seiichi Manyama, Mar 20 2017

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A013661, A109700.

Cf. Sum_{d|n, d=k-1 mod k} d: A000593 (k=2), A078182 (k=3), A050452 (k=4), this sequence (k=5), A284104 (k=6), A284105 (k=7).

Table[Sum[If[Mod[d, 5] == 4, d, 0], {d, Divisors[n]}], {n, 79}] (* Indranil Ghosh, Mar 21 2017 *)
Table[Total[Select[Divisors[n],Mod[#,5]==4&]],{n,80}] (* Harvey P. Dale, Sep 24 2024 *)

for(n=1, 79, print1(sumdiv(n, d, if(Mod(d, 5)==4, d, 0)), ", ")) \\ Indranil Ghosh, Mar 21 2017

from sympy import divisors
def a(n): return sum([d for d in divisors(n) if d%5==4]) # Indranil Ghosh, Mar 21 2017

G.f.: Sum_{k>=1} (5*k - 1)*x^(5*k-1)/(1 - x^(5*k-1)). - Ilya Gutkovskiy, Mar 21 2017

Sum_{k=1..n} a(k) = c * n^2 + O(n*log(n)), where c = Pi^2/60 = 0.164493... (A013661 / 10). - Amiram Eldar, Nov 26 2023

A284105 a(n) = Sum_{d|n, d == 6 (mod 7)} d.

Original entry on oeis.org

0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 6, 13, 0, 0, 0, 0, 6, 0, 20, 0, 0, 0, 6, 0, 13, 27, 0, 0, 6, 0, 0, 0, 34, 0, 6, 0, 0, 13, 20, 41, 6, 0, 0, 0, 0, 0, 54, 0, 0, 0, 13, 0, 33, 55, 0, 0, 0, 0, 26, 0, 62, 0, 0, 13, 6, 0, 34, 69, 0, 0, 6, 0, 0, 0, 76, 0, 19, 0, 20, 27, 41

Offset: 1

Seiichi Manyama, Mar 20 2017

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A109708.
Cf. Sum_{d|n, d == k-1 (mod k)} d: A000593 (k=2), A078182 (k=3), A050452 (k=4), A284103 (k=5), A284104 (k=6), this sequence (k=7).
Cf. Sum_{d|n, d == k (mod 7)} d: A284099 (k=1), A284443 (k=2), A284444 (k=3), A284445 (k=4), A284446 (k=5), this sequence (k=6).

Table[Sum[If[Mod[d,7] == 6,d, 0], {d, Divisors[n]}], {n, 82}] (* Indranil Ghosh, Mar 21 2017 *)

for(n=1, 82, print1(sumdiv(n, d, if(Mod(d,7)==6, d, 0)),", ")) \\ Indranil Ghosh, Mar 21 2017

from sympy import divisors
def a(n): return sum([d for d in divisors(n) if d%7==6]) # Indranil Ghosh, Mar 21 2017

G.f.: Sum_{k>=1} (7*k - 1)*x^(7*k-1)/(1 - x^(7*k-1)). - Ilya Gutkovskiy, Mar 21 2017

Sum_{k=1..n} a(k) = c * n^2 + O(n*log(n)), where c = Pi^2/84 = 0.117495... . - Amiram Eldar, Nov 26 2023

A284104 a(n) = Sum_{d|n, d == 5 (mod 6)} d.

Original entry on oeis.org

0, 0, 0, 0, 5, 0, 0, 0, 0, 5, 11, 0, 0, 0, 5, 0, 17, 0, 0, 5, 0, 11, 23, 0, 5, 0, 0, 0, 29, 5, 0, 0, 11, 17, 40, 0, 0, 0, 0, 5, 41, 0, 0, 11, 5, 23, 47, 0, 0, 5, 17, 0, 53, 0, 16, 0, 0, 29, 59, 5, 0, 0, 0, 0, 70, 11, 0, 17, 23, 40, 71, 0, 0, 0, 5, 0, 88, 0, 0, 5

Offset: 1

Seiichi Manyama, Mar 20 2017

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A086729, A109702.

Cf. Sum_{d|n, d=k-1 mod k} d: A000593 (k=2), A078182 (k=3), A050452 (k=4), A284103 (k=5), this sequence (k=6), A284105 (k=7).

Table[Sum[If[Mod[d, 6] == 5, d, 0], {d, Divisors[n]}], {n, 80}] (* Indranil Ghosh, Mar 21 2017 *)
Table[Total[Select[Divisors[n],Mod[#,6]==5&]],{n,80}] (* Harvey P. Dale, Dec 30 2017 *)

for(n=1, 80, print1(sumdiv(n, d, if(Mod(d,6)==5, d, 0)),", ")) \\ Indranil Ghosh, Mar 21 2017

from sympy import divisors
def a(n): return sum([d for d in divisors(n) if d%6==5]) # Indranil Ghosh, Mar 21 2017

G.f.: Sum_{k>=1} (6*k - 1)*x^(6*k-1)/(1 - x^(6*k-1)). - Ilya Gutkovskiy, Mar 21 2017

Sum_{k=1..n} a(k) = c * n^2 + O(n*log(n)), where c = Pi^2/72 = 0.137077... (A086729). - Amiram Eldar, Nov 26 2023

A293898 Sum of proper divisors of n of the form 3k+2.

Original entry on oeis.org

0, 0, 0, 2, 0, 2, 0, 2, 0, 7, 0, 2, 0, 2, 5, 10, 0, 2, 0, 7, 0, 13, 0, 10, 5, 2, 0, 16, 0, 7, 0, 10, 11, 19, 5, 2, 0, 2, 0, 35, 0, 16, 0, 13, 5, 25, 0, 10, 0, 7, 17, 28, 0, 2, 16, 24, 0, 31, 0, 27, 0, 2, 0, 42, 5, 13, 0, 19, 23, 56, 0, 10, 0, 2, 5, 40, 11, 28, 0, 35, 0, 43, 0, 16, 22, 2, 29, 65, 0, 7, 0, 25, 0, 49, 5, 42, 0, 16, 11, 77, 0, 19, 0, 36, 40

Offset: 1

Antti Karttunen, Nov 06 2017

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences related to sums of divisors.

Cf. A078182, A293896, A293897.

Table[DivisorSum[n, # &, And[Mod[#, 3] == 2, # != n] &], {n, 105}] (* Michael De Vlieger, Nov 08 2017 *)

PARI
```
A293898(n) = sumdiv(n,d,(d
				
```

a(n) = A078182(n) - ([n == 2 (mod 3)]*n).
G.f.: Sum_{k>=1} (3*k-1) * x^(6*k-2) / (1 - x^(3*k-1)). - Ilya Gutkovskiy, Apr 14 2021
Sum_{k=1..n} a(k) = c * n^2 + O(n*log(n)), where c = Pi^2/36  - 1/6 = 0.107489... . - Amiram Eldar, Nov 27 2023

A326400 Expansion of Sum_{k>=1} k * x^(2k) / (1 - x^(3k)).

Original entry on oeis.org

0, 1, 0, 2, 1, 3, 0, 5, 0, 7, 1, 6, 0, 8, 3, 10, 1, 9, 0, 15, 0, 13, 1, 15, 5, 14, 0, 16, 1, 21, 0, 21, 3, 19, 8, 18, 0, 20, 0, 35, 1, 24, 0, 27, 9, 25, 1, 30, 0, 36, 3, 28, 1, 27, 16, 40, 0, 31, 1, 45, 0, 32, 0, 42, 14, 39, 0, 39, 3, 56, 1, 45, 0, 38, 15, 40, 8, 42, 0, 71

Offset: 1

Ilya Gutkovskiy, Sep 11 2019

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A001822, A002131, A016789, A078182, A078708, A326399, A326401.

Cf. A050464, A363900.

nmax = 80; CoefficientList[Series[Sum[k x^(2 k)/(1 - x^(3 k)), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
Table[DivisorSum[n, # &, MemberQ[{2}, Mod[n/#, 3]] &], {n, 1, 80}]

a(n) = Sum_{d|n, n/d==2 (mod 3)} d.

G.f.: Sum_{k>0} x^(3*k-1) / (1 - x^(3*k-1))^2. - Seiichi Manyama, Jun 29 2023

A284315 Expansion of Product_{k>=0} (1 - x^(3*k+2)) in powers of x.

Original entry on oeis.org

1, 0, -1, 0, 0, -1, 0, 1, -1, 0, 1, -1, 0, 2, -1, -1, 2, -1, -1, 3, -1, -2, 3, -1, -3, 4, 0, -4, 4, 0, -5, 5, 1, -7, 5, 2, -8, 6, 4, -10, 5, 5, -12, 6, 8, -14, 5, 10, -16, 5, 14, -19, 3, 17, -21, 2, 22, -23, -1, 26, -26, -3, 33, -28, -8, 38, -30, -12, 46, -32, -19

Offset: 0

Seiichi Manyama, Mar 25 2017

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. Product_{k>=0} (1 - x^(m*k+m-1)): A081362 (m=2), this sequence (m=3), A284316 (m=4), A284317 (m=5).

Cf. A078182, A262928.

CoefficientList[Series[Product[1 - x^(3k + 2), {k, 0, 100}], {x, 0, 100}], x] (* Indranil Ghosh, Mar 25 2017 *)

Vec(prod(k=0, 100, 1 - x^(3*k+2)) + O(x^101)) \\ Indranil Ghosh, Mar 25 2017

a(n) = -(1/n) * Sum_{k=1..n} A078182(k) * a(n-k), a(0) = 1.

A363890 Sum of divisors of 3n-1 of form 3k+2.

Original entry on oeis.org

2, 5, 10, 11, 16, 17, 27, 23, 28, 29, 42, 40, 40, 41, 57, 47, 57, 53, 80, 59, 64, 70, 87, 71, 76, 88, 115, 83, 88, 89, 117, 100, 114, 101, 140, 107, 128, 113, 147, 136, 124, 130, 170, 131, 136, 137, 216, 154, 148, 149, 200, 160, 160, 184, 207, 167, 194, 173, 241, 179, 224, 190, 237

Offset: 1

Seiichi Manyama, Jun 26 2023

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A363514, A363889, A363891.

Cf. A078182, A359211.

a[n_] := DivisorSum[3*n - 1, # &, Mod[#, 3] == 2 &]; Array[a, 100] (* Amiram Eldar, Jun 26 2023 *)

PARI
```
a(n) = sumdiv(3*n-1, d, (d%3==2)*d);
```

a(n) = A078182(3*n-1).

G.f.: Sum_{k>0} (3*k-1) * x^k/(1 - x^(3*k-1)).

cp's OEIS Frontend

A078181 a(n) = Sum_{d|n, d == 1 (mod 3)} d.

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Maple

Mathematica

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A050452 a(n) = Sum_{d|n, d == 3 (mod 4)} d.

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Maple

Mathematica

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A035386 Number of partitions of n into parts congruent to 2 mod 3.

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Maple

Mathematica

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A284103 a(n) = Sum_{d|n, d == 4 (mod 5)} d.

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Mathematica

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A284105 a(n) = Sum_{d|n, d == 6 (mod 7)} d.

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Mathematica

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A284104 a(n) = Sum_{d|n, d == 5 (mod 6)} d.

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Mathematica

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A293898 Sum of proper divisors of n of the form 3k+2.

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A326400 Expansion of Sum_{k>=1} k * x^(2*k) / (1 - x^(3*k)).

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A326400 Expansion of Sum_{k>=1} k * x^(2k) / (1 - x^(3k)).

A363890 Sum of divisors of 3n-1 of form 3k+2.