A078181 -id:A078181 - cp's OEIS frontend

A035382 Number of partitions of n into parts congruent to 1 mod 3.

1, 1, 1, 1, 2, 2, 2, 3, 4, 4, 5, 6, 7, 8, 10, 11, 13, 15, 17, 19, 23, 26, 29, 33, 38, 42, 48, 54, 61, 68, 77, 85, 96, 107, 119, 132, 148, 163, 181, 201, 223, 245, 272, 299, 330, 363, 400, 438, 483, 529, 580, 635, 697, 760, 832, 908, 992, 1081, 1180, 1283, 1399, 1521

Offset: 0

Olivier Gérard

nonn

a(n) = A116373(3*n). - Reinhard Zumkeller, Feb 15 2006

			a(3) = 1 because we have [1,1,1];
a(4) = 2 because we have [1,1,1,1] and [4];
a(9) = 4 because we have [7,1,1], [4,4,1], [4,1,1,1,1,1] and [1,1,1,1,1,1,1,1,1].
1 + x + x^2 + x^3 + 2*x^4 + 2*x^5 + 2*x^6 + 3*x^7 + 4*x^8 + 4*x^9 + ...

Alois P. Heinz and Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Alois P. Heinz)
Joerg Arndt, Matters Computational (The Fxtbook), p. 350.

Cf. A035386, A035451, A261612.

g:= 1/product(1-x^(1+3*j), j=0..50): gser:= series(g, x=0, 64): seq(coeff(gser, x, n), n=0..61); # Emeric Deutsch, Mar 30 2006
# second Maple program
b:= proc(n, i) option remember; `if`(n=0, 1,
      `if`(i<1, 0, b(n, i-3) +`if`(i>n, 0, b(n-i, i))))
    end:
a:= n-> b(n, 3*iquo(n, 3)+1):
seq(a(n), n=0..100);  # Alois P. Heinz, Oct 03 2012

b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, b[n, i-3] + If[i>n, 0, b[n-i, i]]]]; a[n_] := b[n, 3*Quotient[n, 3]+1]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Mar 23 2015, after Alois P. Heinz *)
nmax = 100; poly = ConstantArray[0, nmax + 1]; poly[[1]] = 1; poly[[2]] = -1; Do[If[Mod[k, 3] == 1, 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 + 1;
Table[Count[IntegerPartitions@n, x_ /; SubsetQ[s, x]], {n, 0, nmax}] (* Robert Price, Aug 05 2020 *)

a(n) = 1/n*Sum_{k=1..n} A078181(k)*a(n-k), a(0) = 1.
G.f.: 1/prod(j>=0, 1-x^(1+3*j) ). - Emeric Deutsch, Mar 30 2006
From Joerg Arndt, Oct 02 2012: (Start)
G.f.: sum(n>=0, q^n/prod(k=1..n, 1-q^(3*k)) ); this is the special case of R=1, M=3 of the g.f. sum(n>=0, q^(R*n)/prod(k=1..n, 1-q^(M*k) ) ) for partitions into parts R mod M (where R!=0).
G.f. sum(n>=0, q^(3*n^2-2*n) / prod(k=0..n-1, (1-q^(3*k+3))*(1-q^(3*k+1))) ); this is the special case of R=1, M=3 of the g.f. sum(n>=0, q^(M*n^2+(R-M)*n) / prod(k=0..n-1, (1-q^(M*k+M))*(1-q^(M*k+R))) ) for partitions into parts R mod M (where R!=0). (See Fxtbook link)
(End)
a(n) ~ Gamma(1/3) * exp(sqrt(2*n)*Pi/3) / (2*sqrt(3) * (2*Pi*n)^(2/3)) * (1 + (Pi/72 - 2/(3*Pi)) / sqrt(2*n)). - Vaclav Kotesovec, Feb 26 2015, extended Jan 24 2017
Euler transform of period 3 sequence [ 1, 0, 0, ...]. - Kevin T. Acres, Apr 28 2018

A050449 a(n) = Sum_{d|n, d == 1 (mod 4)} d.

Original entry on oeis.org

1, 1, 1, 1, 6, 1, 1, 1, 10, 6, 1, 1, 14, 1, 6, 1, 18, 10, 1, 6, 22, 1, 1, 1, 31, 14, 10, 1, 30, 6, 1, 1, 34, 18, 6, 10, 38, 1, 14, 6, 42, 22, 1, 1, 60, 1, 1, 1, 50, 31, 18, 14, 54, 10, 6, 1, 58, 30, 1, 6, 62, 1, 31, 1, 84, 34, 1, 18, 70, 6, 1, 10, 74, 38, 31, 1

Offset: 1

N. J. A. Sloane, Dec 23 1999

Not multiplicative: a(3)*a(7) != a(21), for example. - R. J. Mathar, Dec 20 2011

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

Cf. A000593, A050452, A050460, A001826, A035451, A245058.

Cf. Sum_{d|n, d==1 (mod k)} d: A000593 (k=2), A078181 (k=3), this sequence (k=4), A284097 (k=5), A284098 (k=6), A284099 (k=7), A284100 (k=8).

A050449 := proc(n)
        a := 0 ;
        for d in numtheory[divisors](n) do
                if d mod 4 = 1 then
                        a := a+d ;
                end if;
        end do:
        a;
end proc:
seq(A050449(n),n=1..40) ; # R. J. Mathar, Dec 20 2011

a[n_] := DivisorSum[n, Boole[Mod[#, 4] == 1]*#&]; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Jan 30 2018 *)

a(n) = sumdiv(n, d, d*((d % 4) == 1)); \\ Michel Marcus, Jan 30 2018

G.f.: Sum_{n>=0} (4*n+1)*x^(4*n+1)/(1-x^(4*n+1)). - Vladeta Jovovic, Nov 14 2002
a(n) = A000593(n) - A050452(n). - Reinhard Zumkeller, Apr 18 2006
G.f.: Sum_{n >= 1} x^n*(1 + 3*x^(4*n))/(1 - x^(4*n))^2. - Peter Bala, Dec 19 2021
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

More terms from Vladeta Jovovic, Nov 14 2002

More terms from Reinhard Zumkeller, Apr 18 2006

A078182 a(n) = Sum_{d|n, d == 2 (mod 3)} d.

Original entry on oeis.org

0, 2, 0, 2, 5, 2, 0, 10, 0, 7, 11, 2, 0, 16, 5, 10, 17, 2, 0, 27, 0, 13, 23, 10, 5, 28, 0, 16, 29, 7, 0, 42, 11, 19, 40, 2, 0, 40, 0, 35, 41, 16, 0, 57, 5, 25, 47, 10, 0, 57, 17, 28, 53, 2, 16, 80, 0, 31, 59, 27, 0, 64, 0, 42, 70, 13, 0, 87, 23, 56, 71, 10, 0, 76, 5, 40, 88, 28, 0, 115

Offset: 1

Vladeta Jovovic, Nov 21 2002

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

Cf. A001817, A001822, A035386, A078181, A272715, A353908.

A078182 := proc(n)
    a := 0 ;
    for d in numtheory[divisors](n) do
        if modp(d,3) =2 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] == 2 &]; Array[a, 100] (* Giovanni Resta, May 11 2016 *)

a(n) = sumdiv(n, d, d*((d%3) == 2)); \\ Michel Marcus, May 11 2016

G.f.: Sum_{n>=0} (3*n+2)*x^(3*n+2)/(1-x^(3*n+2)).
A078181(n) + a(n) + 3*A000203(n/3) = A000203(n), where A000203 is defined as zero for non-integer arguments. - R. J. Mathar, May 11 2016
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

A284097 a(n) = Sum_{d|n, d == 1 (mod 5)} d.

Original entry on oeis.org

1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 12, 7, 1, 1, 1, 17, 1, 7, 1, 1, 22, 12, 1, 7, 1, 27, 1, 1, 1, 7, 32, 17, 12, 1, 1, 43, 1, 1, 1, 1, 42, 28, 1, 12, 1, 47, 1, 23, 1, 1, 52, 27, 1, 7, 12, 57, 1, 1, 1, 7, 62, 32, 22, 17, 1, 84, 1, 1, 1, 1, 72, 43, 1, 1, 1, 77, 12, 33, 1

Offset: 1

Seiichi Manyama, Mar 20 2017

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

Cf. A013661, A109697.

Cf. Sum_{d|n, d=1 mod k} d: A000593 (k=2), A078181 (k=3), A050449 (k=4), this sequence (k=5), A284098 (k=6), A284099 (k=7), A284100 (k=8).

Table[Sum[If[Mod[d, 5] == 1, d, 0], {d, Divisors[n]}], {n, 80}] (* Indranil Ghosh, Mar 21 2017 *)

for(n=1, 82, print1(sumdiv(n, d, if(Mod(d, 5)==1, 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==1]) # Indranil Ghosh, Mar 21 2017

G.f.: Sum_{k>=0} (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

A284099 a(n) = Sum_{d|n, d == 1 (mod 7)} d.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 9, 1, 1, 1, 1, 1, 1, 16, 9, 1, 1, 1, 1, 1, 23, 1, 9, 1, 1, 1, 1, 30, 16, 1, 9, 1, 1, 1, 37, 1, 1, 1, 9, 1, 1, 44, 23, 16, 1, 1, 9, 1, 51, 1, 1, 1, 1, 1, 9, 58, 30, 1, 16, 1, 1, 1, 73, 1, 23, 1, 1, 1, 1, 72, 45, 1, 1, 16, 1, 1, 79, 1, 9, 1, 1

Offset: 1

Seiichi Manyama, Mar 20 2017

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

Cf. A109703.
Cf. Sum_{d|n, d == 1 (mod k)} d: A000593 (k=2), A078181 (k=3), A050449 (k=4), A284097 (k=5), A284098 (k=6), this sequence (k=7), A284100 (k=8).
Cf. Sum_{d|n, d == k (mod 7)} d: this sequence (k=1), A284443 (k=2), A284444 (k=3), A284445 (k=4), A284446 (k=5), A284105 (k=6).

Table[Sum[If[Mod[d, 7] == 1, d, 0], {d, Divisors[n]}], {n, 82}] (* Indranil Ghosh, Mar 21 2017 *)
Table[DivisorSum[n,#&,Mod[#,7]==1&],{n,90}] (* Harvey P. Dale, Aug 08 2021 *)

for(n=1, 82, print1(sumdiv(n, d, if(Mod(d, 7)==1, 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==1]) # Indranil Ghosh, Mar 21 2017

G.f.: Sum_{k>=0} (7*k + 1)*x^(7*k+1)/(1 - x^(7*k+1)). - Ilya Gutkovskiy, Mar 21 2017
G.f.: Sum_{n >= 1} x^n*(1 + 6*x^(7*n))/(1 - x^(7*n))^2. - Peter Bala, Dec 19 2021
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

A284098 a(n) = Sum_{d|n, d == 1 (mod 6)} d.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 14, 8, 1, 1, 1, 1, 20, 1, 8, 1, 1, 1, 26, 14, 1, 8, 1, 1, 32, 1, 1, 1, 8, 1, 38, 20, 14, 1, 1, 8, 44, 1, 1, 1, 1, 1, 57, 26, 1, 14, 1, 1, 56, 8, 20, 1, 1, 1, 62, 32, 8, 1, 14, 1, 68, 1, 1, 8, 1, 1, 74, 38, 26, 20, 8, 14, 80, 1, 1

Offset: 1

Seiichi Manyama, Mar 20 2017

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

Cf. A086729, A109701.

Cf. Sum_{d|n, d==1 (mod k)} d: A000593 (k=2), A078181 (k=3), A050449 (k=4), A284097 (k=5), this sequence (k=6), A284099 (k=7), A284100 (k=8).

Table[Sum[If[Mod[d, 6] == 1, d, 0], {d, Divisors[n]}], {n, 80}] (* Indranil Ghosh, Mar 21 2017 *)

for(n=1, 82, print1(sumdiv(n, d, if(Mod(d, 6)==1, 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==1]) # Indranil Ghosh, Mar 21 2017

G.f.: Sum_{k>=0} (6*k + 1)*x^(6*k+1)/(1 - x^(6*k+1)). - Ilya Gutkovskiy, Mar 21 2017
G.f.: Sum_{n >= 1} x^n*(1 + 5*x^(6*n))/(1 - x^(6*n))^2. - Peter Bala, Dec 19 2021
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

A284100 a(n) = Sum_{d|n, d == 1 (mod 8)} d.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 10, 1, 1, 1, 1, 1, 1, 1, 18, 10, 1, 1, 1, 1, 1, 1, 26, 1, 10, 1, 1, 1, 1, 1, 34, 18, 1, 10, 1, 1, 1, 1, 42, 1, 1, 1, 10, 1, 1, 1, 50, 26, 18, 1, 1, 10, 1, 1, 58, 1, 1, 1, 1, 1, 10, 1, 66, 34, 1, 18, 1, 1, 1, 10, 74, 1, 26, 1, 1, 1, 1, 1

Offset: 1

Seiichi Manyama, Mar 20 2017

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

Cf. A277090.

Cf. Sum_{d|n, d==1 (mod k)} d: A000593 (k=2), A078181 (k=3), A050449 (k=4), A284097 (k=5), A284098 (k=6), A284099 (k=7), this sequence (k=8).

Table[Sum[If[Mod[d, 8] == 1, d, 0], {d, Divisors[n]}], {n, 80}] (* Indranil Ghosh, Mar 21 2017 *)
Table[Total[Select[Divisors[n],Mod[#,8]==1&]],{n,80}] (* or *) Table[DivisorSum[n,#&,Mod[#,8]==1&],{n,80}] (* Harvey P. Dale, Mar 28 2020 *)

for(n=1, 80, print1(sumdiv(n, d, if(Mod(d, 8)==1, d, 0)), ", ")) \\ Indranil Ghosh, Mar 21 2017

from sympy import divisors
def a(n): return sum([d for d in divisors(n) if d%8==1]) # Indranil Ghosh, Mar 21 2017

G.f.: Sum_{k>=0} (8*k + 1)*x^(8*k+1)/(1 - x^(8*k+1)). - Ilya Gutkovskiy, Mar 21 2017
G.f.: Sum_{n >= 1} x^n*(1 + 7*x^(8*n))/(1 - x^(8*n))^2. - Peter Bala, Dec 19 2021
Sum_{k=1..n} a(k) = c * n^2 + O(n*log(n)), where c = Pi^2/96 = 0.102808... . - Amiram Eldar, Nov 26 2023

A284312 Expansion of Product_{k>=0} (1 - x^(3*k+1)) in powers of x.

Original entry on oeis.org

1, -1, 0, 0, -1, 1, 0, -1, 1, 0, -1, 2, -1, -1, 2, -1, -1, 3, -2, -1, 3, -3, 0, 4, -4, 0, 4, -5, 1, 5, -7, 2, 5, -8, 4, 5, -10, 5, 5, -12, 8, 5, -14, 10, 4, -16, 14, 3, -19, 17, 1, -20, 22, -1, -23, 26, -4, -25, 33, -8, -27, 38, -13, -28, 46, -19, -30, 53, -26, -29

Offset: 0

Seiichi Manyama, Mar 24 2017

sign

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

Cf. Product_{k>=0} (1 - x^(m*k+1)): A081362 (m=2), this sequence (m=3), A284313 (m=4), A284314 (m=5).

Cf. A078181, A261612.

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

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

a(n) = -(1/n)*Sum_{k=1..n} A078181(k)*a(n-k), a(0) = 1.

A293897 Sum of proper divisors of n of the form 3k+1.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 5, 1, 8, 1, 5, 1, 1, 1, 15, 8, 1, 1, 5, 1, 14, 1, 12, 1, 11, 1, 21, 1, 1, 8, 5, 1, 20, 14, 15, 1, 8, 1, 27, 1, 1, 1, 21, 8, 36, 1, 18, 1, 1, 1, 40, 20, 1, 1, 15, 1, 32, 8, 21, 14, 23, 1, 39, 1, 18, 1, 5, 1, 38, 26, 24, 8, 14, 1, 71, 1, 1, 1, 40, 1, 44, 1, 27, 1, 11, 21, 51, 32, 1, 20, 21, 1, 57, 1, 40, 1, 35, 1, 70, 8

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. A078181, A293901, A293895, A293898.

Table[DivisorSum[n, # &, And[Mod[#, 3] == 1, # != n] &], {n, 105}] (* Michael De Vlieger, Nov 08 2017 *)

PARI
```
A293897(n) = sumdiv(n,d,(d
				
```

a(n) = A078181(n) - ([n == 1 (mod 3)]*n).
G.f.: Sum_{k>=1} (3*k-2) * x^(6*k-4) / (1 - x^(3*k-2)). - 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

A326399 Expansion of Sum_{k>=1} k * x^k / (1 - x^(3*k)).

Original entry on oeis.org

1, 2, 3, 5, 5, 6, 8, 10, 9, 11, 11, 15, 14, 16, 15, 21, 17, 18, 20, 27, 24, 23, 23, 30, 26, 28, 27, 40, 29, 33, 32, 42, 33, 35, 40, 45, 38, 40, 42, 55, 41, 48, 44, 57, 45, 47, 47, 63, 57, 57, 51, 70, 53, 54, 56, 80, 60, 59, 59, 81, 62, 64, 72, 85, 70, 69, 68, 87, 69, 88

Offset: 1

Ilya Gutkovskiy, Sep 11 2019

nonn

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

Cf. A001817, A002131, A016777, A050460, A078181, A078708, A326400, A326401.

nmax = 70; CoefficientList[Series[Sum[k x^k/(1 - x^(3 k)), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
Table[DivisorSum[n, # &, MemberQ[{1}, Mod[n/#, 3]] &], {n, 1, 70}]

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

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

cp's OEIS Frontend

A035382 Number of partitions of n into parts congruent to 1 mod 3.

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A050449 a(n) = Sum_{d|n, d == 1 (mod 4)} d.

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A078182 a(n) = Sum_{d|n, d == 2 (mod 3)} d.

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A284097 a(n) = Sum_{d|n, d == 1 (mod 5)} d.

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A284099 a(n) = Sum_{d|n, d == 1 (mod 7)} d.

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A284098 a(n) = Sum_{d|n, d == 1 (mod 6)} d.

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A284100 a(n) = Sum_{d|n, d == 1 (mod 8)} d.

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