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.

A178143 Sum of squares d^2 over the divisors d=2 and/or d=3 of n.

Original entry on oeis.org

0, 4, 9, 4, 0, 13, 0, 4, 9, 4, 0, 13, 0, 4, 9, 4, 0, 13, 0, 4, 9, 4, 0, 13, 0, 4, 9, 4, 0, 13, 0, 4, 9, 4, 0, 13, 0, 4, 9, 4, 0, 13, 0, 4, 9, 4, 0, 13, 0, 4, 9, 4, 0, 13, 0, 4, 9, 4, 0, 13, 0, 4, 9, 4, 0, 13, 0, 4, 9, 4, 0, 13, 0, 4, 9, 4, 0, 13, 0, 4, 9, 4
Offset: 1

Views

Author

Vladimir Shevelev, May 21 2010

Keywords

Comments

Period 6: repeat [0, 4, 9, 4, 0, 13]. - Wesley Ivan Hurt, Jul 05 2016

Examples

			a(1)=0, a(2)=2^2=4 since 2|2, a(3)=3^2=9 since 3|3, a(4)=2^2=4 since 2|4.

Links

Crossrefs

Cf. A010675, A021115, A098002, A178142.

Programs

Formula

a(n) = Sum_{d|n, d=2 and/or d=3} d^2.
a(n) = -a(n-1) + a(n-3) + a(n-4) for n>4.
G.f.: x*(4+13*x+13*x^2) / ( (1-x)*(1+x)*(1+x+x^2) ).
a(n+6) = a(n).
a(n) = A010675(n) + A021115(n). [R. J. Mathar, May 28 2010]
a(n) = 4 * (1 + floor(n/2) - ceiling(n/2)) + 9 * (1 + floor(n/3) - ceiling(n/3)). - Wesley Ivan Hurt, May 20 2013
a(n) = 5 + 2*cos(n*Pi) + 6*cos(2*n*Pi/3). - Wesley Ivan Hurt, Jul 05 2016

Extensions

Replaced recurrence by a shorter one; added keyword:less - R. J. Mathar, May 28 2010

A347156 Sum of squares of distinct prime divisors of n that are < sqrt(n).

Original entry on oeis.org

0, 0, 0, 0, 0, 4, 0, 4, 0, 4, 0, 13, 0, 4, 9, 4, 0, 13, 0, 4, 9, 4, 0, 13, 0, 4, 9, 4, 0, 38, 0, 4, 9, 4, 25, 13, 0, 4, 9, 29, 0, 13, 0, 4, 34, 4, 0, 13, 0, 29, 9, 4, 0, 13, 25, 53, 9, 4, 0, 38, 0, 4, 58, 4, 25, 13, 0, 4, 9, 78, 0, 13, 0, 4, 34, 4, 49, 13, 0, 29
Offset: 1

Views

Author

Ilya Gutkovskiy, Aug 20 2021

Keywords

Crossrefs

Cf. A001157, A005063, A005066, A098002, A333806, A333808, A339353, A347157, A347158.

Programs

  • Mathematica
    Table[DivisorSum[n, #^2 &, # < Sqrt[n] && PrimeQ[#] &], {n, 1, 80}]
nmax = 80; CoefficientList[Series[Sum[Prime[k]^2 x^(Prime[k] (Prime[k] + 1))/(1 - x^Prime[k]), {k, 1, nmax}], {x, 0, nmax}], x] // Rest

Formula

G.f.: Sum_{k>=1} prime(k)^2 * x^(prime(k)*(prime(k) + 1)) / (1 - x^prime(k)).

A347159 Sum of cubes of distinct prime divisors of n that are <= sqrt(n).

Original entry on oeis.org

0, 0, 0, 8, 0, 8, 0, 8, 27, 8, 0, 35, 0, 8, 27, 8, 0, 35, 0, 8, 27, 8, 0, 35, 125, 8, 27, 8, 0, 160, 0, 8, 27, 8, 125, 35, 0, 8, 27, 133, 0, 35, 0, 8, 152, 8, 0, 35, 343, 133, 27, 8, 0, 35, 125, 351, 27, 8, 0, 160, 0, 8, 370, 8, 125, 35, 0, 8, 27, 476, 0, 35, 0, 8, 152
Offset: 1

Views

Author

Ilya Gutkovskiy, Aug 20 2021

Keywords

Crossrefs

Cf. A001158, A005064, A005067, A063962, A097974, A098002, A280375, A347157, A347160.

Programs

  • Mathematica
    Table[DivisorSum[n, #^3 &, # <= Sqrt[n] && PrimeQ[#] &], {n, 1, 75}]
nmax = 75; CoefficientList[Series[Sum[Prime[k]^3 x^(Prime[k]^2)/(1 - x^Prime[k]), {k, 1, nmax}], {x, 0, nmax}], x] // Rest

Formula

G.f.: Sum_{k>=1} prime(k)^3 * x^(prime(k)^2) / (1 - x^prime(k)).

A347160 Sum of 4th powers of distinct prime divisors of n that are <= sqrt(n).

Original entry on oeis.org

0, 0, 0, 16, 0, 16, 0, 16, 81, 16, 0, 97, 0, 16, 81, 16, 0, 97, 0, 16, 81, 16, 0, 97, 625, 16, 81, 16, 0, 722, 0, 16, 81, 16, 625, 97, 0, 16, 81, 641, 0, 97, 0, 16, 706, 16, 0, 97, 2401, 641, 81, 16, 0, 97, 625, 2417, 81, 16, 0, 722, 0, 16, 2482, 16, 625, 97, 0, 16, 81, 3042
Offset: 1

Views

Author

Ilya Gutkovskiy, Aug 20 2021

Keywords

Crossrefs

Cf. A001159, A005065, A005068, A063962, A097974, A098002, A347143, A347158, A347159.

Programs

  • Mathematica
    Table[DivisorSum[n, #^4 &, # <= Sqrt[n] && PrimeQ[#] &], {n, 1, 70}]
nmax = 70; CoefficientList[Series[Sum[Prime[k]^4 x^(Prime[k]^2)/(1 - x^Prime[k]), {k, 1, nmax}], {x, 0, nmax}], x] // Rest

Formula

G.f.: Sum_{k>=1} prime(k)^4 * x^(prime(k)^2) / (1 - x^prime(k)).

A178147 Sum of squares d^2 of distinct divisors of n, d in {2, 3, 5}.

Original entry on oeis.org

0, 4, 9, 4, 25, 13, 0, 4, 9, 29, 0, 13, 0, 4, 34, 4, 0, 13, 0, 29, 9, 4, 0, 13, 25, 4, 9, 4, 0, 38, 0, 4, 9, 4, 25, 13, 0, 4, 9, 29, 0, 13, 0, 4, 34, 4, 0, 13, 0, 29, 9, 4, 0, 13, 25, 4, 9, 4, 0, 38, 0, 4, 9, 4, 25, 13, 0, 4, 9, 29, 0, 13, 0, 4, 34, 4, 0, 13, 0
Offset: 1

Views

Author

Vladimir Shevelev, May 21 2010, May 23 2010

Keywords

Comments

The sequence is periodic with period {0 4 9 4 25 13 0 4 9 29 0 13 0 4 34 4 0 13 0 29 9 4 0 13 25 4 9 4 0 38} of length 30.
A generalization: let B={b_1,...,b_t} be a set of t positive (not necessarily distinct) integers and m>=0 an integer.
For m>=0, let A(n)=Sum d^m over divisors d of n which are elements of B (with the multiplicities as in B). Calculating directly values of
A(b_i),A(b_i+b_j),A(b_i+b_j+b_k),...,
A(b_1+...+b_t), for the other values of A(n) we have the recursion:
A(n)=Sum{1<=i<=t}A(n-b_i)- Sum{1<=i

Links

Crossrefs

Cf. A005063, A098002, A178142-A178144, A178146.

Formula

a(n)= a(n-2) +a(n-3) -a(n-7)- a(n-8) +a(n-10), n>10.
By the comment, up to 10 it is sufficient to
calculate directly only values a(2)=4, a(3)=9, a(5)=25, a(7)=0, a(8)=4, a(10)=29.
For other n's we can use the recursion, accepting formally a(n)=0 for n<0. So a(1)=0; a(4)=a(2)+a(1)=4;a(6)=a(4)+a(3)=4+9=13,
a(9)=a(7)+a(6)-a(2)-a(1)=0+13-4+0=9.
a(n) = -2*a(n-1) -2*a(n-2) -a(n-3) +a(n-5) +2*a(n-6) +2*a(n-7) +a(n-8). - R. J. Mathar, Jul 13 2010
G.f. -x^2*(4+17*x+30*x^2+55*x^3+80*x^4+38*x^6+76*x^5) / ( (x-1)*(1+x)*(1+x+x^2)*(x^4+x^3+x^2+x+1) ). - R. J. Mathar, Dec 17 2012
Showing 1-5 of 5 results.

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