A284099 -id:A284099 - cp's OEIS frontend

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

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

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

Original entry on oeis.org

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

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

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

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

A284443 a(n) = Sum_{d|n, d == 2 (mod 7)} d.

Original entry on oeis.org

0, 2, 0, 2, 0, 2, 0, 2, 9, 2, 0, 2, 0, 2, 0, 18, 0, 11, 0, 2, 0, 2, 23, 2, 0, 2, 9, 2, 0, 32, 0, 18, 0, 2, 0, 11, 37, 2, 0, 2, 0, 2, 0, 46, 9, 25, 0, 18, 0, 2, 51, 2, 0, 11, 0, 2, 0, 60, 0, 32, 0, 2, 9, 18, 65, 2, 0, 2, 23, 2, 0, 83, 0, 39, 0, 2, 0, 2, 79, 18, 9

Offset: 1

Seiichi Manyama, Mar 27 2017

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

Cf. A109704.

Cf. Sum_{d|n, d == k (mod 7)} d: A284099 (k=1), this sequence (k=2), A284444 (k=3), A284445 (k=4), A284446 (k=5), A284105 (k=6).

Table[Sum[If[Mod[d, 7] == 2, d, 0], {d, Divisors[n]}], {n, 80}] (* Indranil Ghosh, Mar 27 2017 *)

a(n) = sumdiv(n, d, d*((d % 7) == 2)); \\ Amiram Eldar, Nov 26 2023

G.f.: Sum_{k>=0} (7*k + 2)*x^(7*k+2)/(1 - x^(7*k+2)). - Ilya Gutkovskiy, Mar 28 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

A284444 a(n) = Sum_{d|n, d == 3 (mod 7)} d.

Original entry on oeis.org

0, 0, 3, 0, 0, 3, 0, 0, 3, 10, 0, 3, 0, 0, 3, 0, 17, 3, 0, 10, 3, 0, 0, 27, 0, 0, 3, 0, 0, 13, 31, 0, 3, 17, 0, 3, 0, 38, 3, 10, 0, 3, 0, 0, 48, 0, 0, 27, 0, 10, 20, 52, 0, 3, 0, 0, 3, 0, 59, 13, 0, 31, 3, 0, 0, 69, 0, 17, 3, 10, 0, 27, 73, 0, 3, 38, 0, 3, 0, 90, 3

Offset: 1

Seiichi Manyama, Mar 27 2017

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

Cf. A109705.

Cf. Sum_{d|n, d == k (mod 7)} d: A284099 (k=1), A284443 (k=2), this sequence (k=3), A284445 (k=4), A284446 (k=5), A284105 (k=6).

Table[Sum[If[Mod[d, 7] == 3, d, 0], {d, Divisors[n]}], {n, 80}] (* Indranil Ghosh, Mar 27 2017 *)

a(n) = sumdiv(n, d, d*((d % 7) == 3)); \\ Amiram Eldar, Nov 26 2023

G.f.: Sum_{k>=0} (7*k + 3)*x^(7*k+3)/(1 - x^(7*k+3)). - Ilya Gutkovskiy, Mar 28 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

A284445 a(n) = Sum_{d|n, d == 4 (mod 7)} d.

Original entry on oeis.org

0, 0, 0, 4, 0, 0, 0, 4, 0, 0, 11, 4, 0, 0, 0, 4, 0, 18, 0, 4, 0, 11, 0, 4, 25, 0, 0, 4, 0, 0, 0, 36, 11, 0, 0, 22, 0, 0, 39, 4, 0, 0, 0, 15, 0, 46, 0, 4, 0, 25, 0, 4, 53, 18, 11, 4, 0, 0, 0, 64, 0, 0, 0, 36, 0, 11, 67, 4, 0, 0, 0, 22, 0, 74, 25, 4, 11, 39, 0, 4, 81

Offset: 1

Seiichi Manyama, Mar 27 2017

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

Cf. A109706.

Cf. Sum_{d|n, d == k (mod 7)} d: A284099 (k=1), A284443 (k=2), A284444 (k=3), this sequence (k=4), A284446 (k=5), A284105 (k=6).

Table[Sum[If[Mod[d, 7] == 4, d, 0], {d, Divisors[n]}], {n, 80}] (* Indranil Ghosh, Mar 27 2017 *)

a(n) = sumdiv(n, d, d*((d % 7) == 4)); \\ Amiram Eldar, Nov 26 2023

G.f.: Sum_{k>=0} (7*k + 4)*x^(7*k+4)/(1 - x^(7*k+4)). - Ilya Gutkovskiy, Mar 28 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

A284446 a(n) = Sum_{d|n, d == 5 (mod 7)} d.

Original entry on oeis.org

0, 0, 0, 0, 5, 0, 0, 0, 0, 5, 0, 12, 0, 0, 5, 0, 0, 0, 19, 5, 0, 0, 0, 12, 5, 26, 0, 0, 0, 5, 0, 0, 33, 0, 5, 12, 0, 19, 0, 45, 0, 0, 0, 0, 5, 0, 47, 12, 0, 5, 0, 26, 0, 54, 5, 0, 19, 0, 0, 17, 61, 0, 0, 0, 5, 33, 0, 68, 0, 5, 0, 12, 0, 0, 80, 19, 0, 26, 0, 45, 0, 82

Offset: 1

Seiichi Manyama, Mar 27 2017

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A109707.

Cf. Sum_{d|n, d == k (mod 7)} d: A284099 (k=1), A284443 (k=2), A284444 (k=3), A284445 (k=4), this sequence (k=5), A284105 (k=6).

f:= n -> convert(select(t -> t mod 7 = 5, numtheory:-divisors(n)),`+`):
map(f, [$1..1000]); # Robert Israel, Mar 27 2017

Table[DivisorSum[n, # &, Mod[#, 7] == 5 &], {n, 82}] (* Giovanni Resta, Mar 27 2017 *)

for(n=1, 82, print1(sumdiv(n, d, if(Mod(d, 7)==5, d, 0)), ", ")) \\ Indranil Ghosh, Mar 27 2017

G.f.: Sum_{k>=0} (5+7k) x^(5+7k)/(1-x^(5+7k)). - Robert Israel, Mar 27 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

A109703 Number of partitions of n into parts each equal to 1 mod 7.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 3, 4, 4, 4, 4, 4, 4, 5, 6, 7, 7, 7, 7, 7, 8, 10, 11, 12, 12, 12, 12, 13, 15, 17, 18, 19, 19, 19, 20, 23, 26, 28, 29, 30, 30, 31, 34, 38, 41, 43, 44, 45, 46, 50, 55, 60, 63, 65, 66, 68, 72, 79, 85, 90, 93, 95, 97, 103, 111, 120, 127, 132, 135

Offset: 0

Erich Friedman, Aug 07 2005

nonn

			a(15)=3 because we have 15=8+1+1+1+1+1+1+1=1+1+1+1+1+1+1+1+1+1+1+1+1+1+1.

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
Vaclav Kotesovec, Graph - The asymptotic ratio

Cf. A284099.

Cf. similar sequences of number of partitions of n into parts congruent to 1 mod m: A000009 (m=2), A035382 (m=3), A035451 (m=4), A109697 (m=5), A109701 (m=6), this sequence (m=7), A277090 (m=8).

g:=1/product(1-x^(1+7*j),j=0..20): gser:=series(g,x=0,80): seq(coeff(gser,x,n),n=0..77); # Emeric Deutsch, Apr 14 2006

nmax=100; CoefficientList[Series[Product[1/(1-x^(7*k+1)),{k, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Feb 27 2015 *)

G.f.: 1/product(1-x^(1+7j), j=0..infinity). - Emeric Deutsch, Apr 14 2006
a(n) ~ Gamma(1/7) * exp(Pi*sqrt(2*n/21)) / (2^(11/7) * 3^(1/14) * 7^(3/7) * Pi^(6/7) * n^(4/7)) * (1 - (2*sqrt(6/7)/(7*Pi) + 13*Pi/(168*sqrt(42))) / sqrt(n)). - Vaclav Kotesovec, Feb 27 2015, extended Jan 24 2017
a(n) = (1/n)*Sum_{k=1..n} A284099(k)*a(n-k), a(0) = 1. - Seiichi Manyama, Mar 20 2017
G.f.: Sum_{k>=0} x^k / Product_{j=1..k} (1 - x^(7*j)). - Ilya Gutkovskiy, Jul 17 2019

Changed offset to 0 and added a(0)=1 by Vaclav Kotesovec, Feb 27 2015

cp's OEIS Frontend

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

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

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

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

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

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A284443 a(n) = Sum_{d|n, d == 2 (mod 7)} d.

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A284444 a(n) = Sum_{d|n, d == 3 (mod 7)} d.

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A284445 a(n) = Sum_{d|n, d == 4 (mod 7)} d.

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