cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-9 of 9 results.

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

Views

Author

N. J. A. Sloane, Dec 23 1999

Keywords

Comments

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

Links

Crossrefs

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).

Programs

  • Maple
    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
  • Mathematica
    a[n_] := DivisorSum[n, Boole[Mod[#, 4] == 1]*#&]; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Jan 30 2018 *)
  • PARI
    a(n) = sumdiv(n, d, d*((d % 4) == 1)); \\ Michel Marcus, Jan 30 2018

Formula

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

Extensions

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

Views

Author

Vladeta Jovovic, Nov 21 2002

Keywords

Links

Crossrefs

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).

Programs

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

Formula

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

Views

Author

Seiichi Manyama, Mar 20 2017

Keywords

Links

Crossrefs

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).

Programs

  • Mathematica
    Table[Sum[If[Mod[d, 5] == 1, d, 0], {d, Divisors[n]}], {n, 80}] (* Indranil Ghosh, Mar 21 2017 *)
  • PARI
    for(n=1, 82, print1(sumdiv(n, d, if(Mod(d, 5)==1, d, 0)), ", ")) \\ Indranil Ghosh, Mar 21 2017
  • Python
    from sympy import divisors
def a(n): return sum([d for d in divisors(n) if d%5==1]) # Indranil Ghosh, Mar 21 2017

Formula

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

Views

Author

Seiichi Manyama, Mar 20 2017

Keywords

Links

Crossrefs

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).

Programs

  • Mathematica
    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 *)
  • PARI
    for(n=1, 82, print1(sumdiv(n, d, if(Mod(d, 7)==1, d, 0)), ", ")) \\ Indranil Ghosh, Mar 21 2017
  • Python
    from sympy import divisors
def a(n): return sum([d for d in divisors(n) if d%7==1]) # Indranil Ghosh, Mar 21 2017

Formula

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

A186099 Sum of divisors of n congruent to 1 or 5 mod 6.

Original entry on oeis.org

1, 1, 1, 1, 6, 1, 8, 1, 1, 6, 12, 1, 14, 8, 6, 1, 18, 1, 20, 6, 8, 12, 24, 1, 31, 14, 1, 8, 30, 6, 32, 1, 12, 18, 48, 1, 38, 20, 14, 6, 42, 8, 44, 12, 6, 24, 48, 1, 57, 31, 18, 14, 54, 1, 72, 8, 20, 30, 60, 6, 62, 32, 8, 1, 84, 12, 68, 18, 24, 48, 72, 1, 74, 38, 31, 20, 96, 14, 80, 6
Offset: 1

Views

Author

Michael Somos, Feb 12 2011

Keywords

Comments

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

Examples

			G.f.: x + x^2 + x^3 + x^4 + 6*x^5 + x^6 + 8*x^7 + x^8 + x^9 + 6*x^10 + 12*x^11 +...
L.g.f.: L(x) = x + x^2/2 + x^3/3 + x^4/4 + 6*x^5/5 + x^6/6 + 8*x^7/7 + x^8/8 +...
where exp(L(x)) = 1 + x + x^2 + x^3 + x^4 + 2*x^5 + 2*x^6 + 3*x^7 + 3*x^8 + 3*x^9 +...+ A003105(n)*x^n +...

Links

Crossrefs

Cf. A000593, A046913, A113957, A116073, A003105, A284098, A284104, A353908.
Cf. A004016, A005882, A005928.

Programs

  • Mathematica
    Table[Total[Select[Divisors[n],MemberQ[{1,5},Mod[#,6]]&]],{n,0,100}]  (* Harvey P. Dale, Feb 24 2011 *)
a[ n_] := If[ n < 1, 0, DivisorSum[n, If[ 1 == GCD[#, 6], #, 0] &]]; (* Michael Somos, Jun 27 2017 *)
a[ n_] := If[n < 1, 0, Times @@ (Which[# < 5, 1, True, (#^(#2 + 1) - 1) / (# - 1)] & @@@ FactorInteger[n])]; (* Michael Somos, Jun 27 2017 *)
  • PARI
    {a(n) = sumdiv( n, d, d * (1 == gcd(d, 6) ))};
  • PARI
    {a(n) = direuler( p=2, n, 1 / (1 - X) / (1 - (p>3) * p * X)) [n]};
  • PARI
    a(n)=sigma(n/2^valuation(n,2)/3^valuation(n,3)) \\ Charles R Greathouse IV, Dec 07 2011
  • PARI
    {S(n,x)=sumdiv(n,d,d*(1-x^d)^(n/d))}
{a(n)=n*polcoeff(sum(k=1,n,S(k,x)*x^k/k)+x*O(x^n),n)}
for(n=1,80,print1(a(n),", "))
/* Paul D. Hanna, Feb 17 2013 */

Formula

Expansion of (1 + a(x)^2 - 2*a(x^2)^2) / 12 in powers of x where a() is a cubic AGM function.
a(n) is multiplicative with a(2^e) = a(3^e) = 1, a(p^e) = (p^(e+1) - 1) / (p - 1) if p>3.
Equals the logarithmic derivative of A003105, where A003105(n) = number of partitions of n into parts 6*n+1 or 6*n-1. - Paul D. Hanna, Feb 17 2013
L.g.f.: Sum_{n>=1} a(n)*x^n/n = Sum_{n>=1} S(n,x)*x^n/n where S(n,x) = Sum_{d|n} d*(1-x^d)^(n/d). - Paul D. Hanna, Feb 17 2013
a(n) = A284098(n) + A284104(n). - Seiichi Manyama, Mar 24 2017
G.f.: Sum_{n >= 1} x^n*(x^(10*n) + 5*x^(6*n) + 5*x^(4*n) + 1)/(1 - x^(6*n))^2. - Peter Bala, Dec 19 2021
From Amiram Eldar, Dec 30 2022: (Start)
Dirichlet g.f.: zeta(s)*zeta(s-1)*(1-2/2^s)*(1-3/3^s).
Sum_{k=1..n} a(k) ~ c*n^2, where c = Pi^2/36 = 0.274155... (A353908). (End)

A109701 Number of partitions of n into parts each equal to 1 mod 6.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 4, 4, 4, 4, 4, 5, 6, 7, 7, 7, 7, 8, 10, 11, 12, 12, 12, 13, 15, 17, 18, 19, 19, 20, 23, 26, 28, 29, 30, 31, 34, 38, 41, 43, 44, 46, 50, 55, 60, 63, 65, 67, 72, 79, 85, 90, 93, 96, 102, 111, 120, 127, 132, 136, 143, 154, 166, 176, 183, 189, 198
Offset: 0

Views

Author

Erich Friedman, Aug 07 2005

Keywords

Comments

Euler transform of period 6 sequence [ 1, 0, 0, 0, 0, 0, ...]. - Kevin T. Acres, Apr 28 2018

Examples

			a(10)=2 since 10 = 7+1+1+1 = 1+1+1+1+1+1+1+1+1+1

Links

Crossrefs

Cf. A284098.
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), this sequence (m=6), A109703 (m=7), A277090 (m=8).

Programs

  • Maple
    g:=1/product(1-x^(1+6*j),j=0..20): gser:=series(g,x=0,77): seq(coeff(gser,x,n),n=0..74); # Emeric Deutsch, Apr 14 2006
  • Mathematica
    nmax=100; CoefficientList[Series[Product[1/(1-x^(6*k+1)),{k, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Feb 27 2015 *)

Formula

G.f.: 1/Product_{j >= 0} (1-x^(1+6j)). - Emeric Deutsch, Apr 14 2006
a(n) ~ Gamma(1/6) * exp(Pi*sqrt(n)/3) / (4 * sqrt(3) * Pi^(5/6) * n^(7/12)) * (1 - (7/(24*Pi) + Pi/144) / sqrt(n)). - Vaclav Kotesovec, Feb 27 2015, extended Jan 24 2017
a(n) = (1/n)*Sum_{k=1..n} A284098(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^(6*j)). - Ilya Gutkovskiy, Jul 17 2019

Extensions

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

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

Views

Author

Seiichi Manyama, Mar 20 2017

Keywords

Links

Crossrefs

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).

Programs

  • Mathematica
    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 *)
  • PARI
    for(n=1, 80, print1(sumdiv(n, d, if(Mod(d, 8)==1, d, 0)), ", ")) \\ Indranil Ghosh, Mar 21 2017
  • Python
    from sympy import divisors
def a(n): return sum([d for d in divisors(n) if d%8==1]) # Indranil Ghosh, Mar 21 2017

Formula

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

A284585 Expansion of Product_{k>=0} (1 - x^(6*k+1)) in powers of x.

Original entry on oeis.org

1, -1, 0, 0, 0, 0, 0, -1, 1, 0, 0, 0, 0, -1, 1, 0, 0, 0, 0, -1, 2, -1, 0, 0, 0, -1, 2, -1, 0, 0, 0, -1, 3, -2, 0, 0, 0, -1, 3, -3, 1, 0, 0, -1, 4, -4, 1, 0, 0, -1, 4, -5, 2, 0, 0, -1, 5, -7, 3, 0, 0, -1, 5, -8, 5, -1, 0, -1, 6, -10, 6, -1, 0, -1, 6, -12, 9, -2, 0, -1
Offset: 0

Views

Author

Seiichi Manyama, Mar 29 2017

Keywords

Links

Crossrefs

Cf. Product_{k>=0} (1 - x^(6*k+m)): this sequence (m=1), A284586 (m=5).
Cf. A280456.

Programs

  • Maple
    N:= 100: # to get a(0)..a(N)
V:= Vector(N+1):
V[1]:= 1:
for k from 0 to floor((N-1)/6) do
V[6*k+2..N+1]:= V[6*k+2..N+1]+V[1..N-6*k]
od:
seq((-1)^n*V[n+1],n=0..N); # Robert Israel, Apr 09 2017
  • Mathematica
    CoefficientList[Series[Product[1 - x^(6*k+1), {k, 0, 79}], {x, 0, 79}], x] (* Indranil Ghosh, Mar 29 2017 *)

Formula

a(n) = -(1/n)*Sum_{k=1..n} A284098(k)*a(n-k), a(0) = 1.
a(n) = (-1)^n * A280456(n). - Robert Israel, Apr 09 2017

A357912 a(n) = Sum_{d|n, d==1 (mod 11)} d.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 24, 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 35, 1, 13, 1, 1, 1, 1, 1, 1, 1, 1, 46, 24, 1, 13, 1, 1, 1, 1, 1, 1, 1, 57, 1, 1, 1, 13, 1, 1, 1, 1, 1, 1, 68, 35, 24, 1, 1, 13, 1, 1, 1, 1, 1, 79, 1, 1, 1, 1, 1, 13, 1
Offset: 1

Views

Author

Seiichi Manyama, Jan 17 2023

Keywords

Links

Crossrefs

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), A284100 (k=8), this sequence (k=11).
Cf. A357911.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, # &, Mod[#, 11] == 1 &]; Array[a, 100] (* Amiram Eldar, Aug 09 2023 *)
  • PARI
    a(n) = sumdiv(n, d, (Mod(d, 11)==1)*d);
  • PARI
    my(N=100, x='x+O('x^N)); Vec(sum(k=0, N, (11*k+1)*x^(11*k+1)/(1-x^(11*k+1))))

Formula

G.f.: Sum_{k>=0} (11*k+1) * x^(11*k+1)/(1 - x^(11*k+1)).
Showing 1-9 of 9 results.

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