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 1-5 of 5 results.

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

Views

Author

Seiichi Manyama, Mar 20 2017

Keywords

Links

Crossrefs

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).

Programs

  • Mathematica
    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 *)
  • PARI
    for(n=1, 79, print1(sumdiv(n, d, if(Mod(d, 5)==4, 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==4]) # Indranil Ghosh, Mar 21 2017

Formula

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

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)

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

Views

Author

Seiichi Manyama, Mar 20 2017

Keywords

Links

Crossrefs

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).

Programs

  • Mathematica
    Table[Sum[If[Mod[d,7] == 6,d, 0], {d, Divisors[n]}], {n, 82}] (* Indranil Ghosh, Mar 21 2017 *)
  • PARI
    for(n=1, 82, print1(sumdiv(n, d, if(Mod(d,7)==6, 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==6]) # Indranil Ghosh, Mar 21 2017

Formula

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

A109702 Number of partitions of n into parts each equal to 5 mod 6.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 1, 1, 2, 1, 0, 1, 1, 2, 2, 1, 1, 1, 2, 3, 3, 2, 1, 2, 3, 4, 4, 2, 2, 3, 5, 6, 5, 3, 3, 5, 7, 8, 6, 4, 5, 8, 10, 10, 8, 6, 8, 11, 13, 13, 10, 9, 12, 15, 18, 17, 14, 13, 16, 21, 23, 22, 18, 18, 23, 28, 31, 28, 24, 25, 31, 38, 39, 36, 32, 34
Offset: 0

Views

Author

Erich Friedman, Aug 07 2005

Keywords

Examples

			a(40)=4 since 40 = 35+5 = 29+11 = 23+17 = 5+5+5+5+5+5+5+5.

Links

Crossrefs

Cf. A284104.
Cf. similar sequences of number of partitions of n into parts congruent to m-1 mod m: A000009 (m=2), A035386 (m=3), A035462 (m=4), A109700 (m=5), this sequence (m=6), A109708 (m=7).

Programs

Formula

G.f.: 1/product(1-x^(5+6j),j=0..infinity). - Emeric Deutsch, Apr 14 2006
a(n) ~ Gamma(5/6) * exp(Pi*sqrt(n)/3) / (4 * sqrt(3) * Pi^(1/6) * n^(11/12)) * (1 - (55/(24*Pi) + Pi/144) / sqrt(n)). - Vaclav Kotesovec, Feb 27 2015, extended Jan 24 2017
a(n) = (1/n)*Sum_{k=1..n} A284104(k)*a(n-k), a(0) = 1. - Seiichi Manyama, Mar 20 2017
Euler transform of period 6 sequence [ 0, 0, 0, 0, 1, 0, ...]. - Kevin T. Acres, Apr 28 2018

Extensions

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

A284586 Expansion of Product_{k>=0} (1 - x^(6*k+5)) in powers of x.

Original entry on oeis.org

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

Views

Author

Seiichi Manyama, Mar 29 2017

Keywords

Crossrefs

Cf. Product_{k>=0} (1 - x^(6*k+m)): A284585 (m=1), this sequence (m=5).
Cf. A281244.

Programs

  • Mathematica
    CoefficientList[Series[Product[1 - x^(6*k+5), {k, 0, 80}], {x, 0, 80}], x] (* Indranil Ghosh, Mar 29 2017 *)

Formula

a(n) = -(1/n)*Sum_{k=1..n} A284104(k)*a(n-k), a(0) = 1.
Showing 1-5 of 5 results.

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