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.

A133180 a(n) = (Sum_{k=1..A047380(n)} k^6) / (Sum_{k=1..A047380(n)} k^2).

Original entry on oeis.org

1, 13, 163, 373, 2191, 3433, 7411, 10363, 24583, 31591, 49981, 61723, 109513, 130351, 180793, 210901, 324013, 370273, 477463, 539041, 759811, 846613, 1042303, 1151983, 1533331, 1679323, 2002321, 2180263, 2785693, 3013051, 3509221
Offset: 1

Views

Author

Artur Jasinski, Oct 10 2007

Keywords

Comments

A000540(n) is divisible by A000330(n) if and only if n is congruent to {1,2,4,5} mod 7 (see A047380).
This sequence is the union of A134158 and A134159 and A134160 and A134161.

Crossrefs

Cf. A000330, A000540, A047380, A119617, A134153, A134154, A134158, A134159, A134160, A134161.

Programs

  • Mathematica
    a = {}; Do[j = Sum[k^6, {k, 1, n}]/Sum[k^2, {k, 1, n}]; If[IntegerQ[j], AppendTo[a, j]], {n, 1, 100}] ; a (*Artur Jasinski*)
Select[Table[Sum[k^6,{k,n}]/Sum[k^2,{k,n}],{n,100}],IntegerQ] (* Harvey P. Dale, Nov 26 2019 *)

Formula

a(n) = A000540(A047380(n)) / A000330(A047380(n)). - Jason Yuen, Sep 23 2024

Extensions

Offset corrected by Jason Yuen, Sep 23 2024

A134158 a(n) = 1 + 27n + 252n^2 + 882n^3 + 1029n^4.

Original entry on oeis.org

1, 2191, 24583, 109513, 324013, 759811, 1533331, 2785693, 4682713, 7414903, 11197471, 16270321, 22898053, 31369963, 42000043, 55126981, 71114161, 90349663, 113246263, 140241433, 171797341, 208400851, 250563523, 298821613, 353736073, 415892551, 485901391
Offset: 0

Views

Author

Artur Jasinski, Oct 10 2007

Keywords

Comments

A000540(n) is divisible by A000330(n) if and only if n is congruent to {1,2,4,5} mod 7 (see A047380).
This sequence is the case when n is congruent to 1 mod 7.
A134159 is the case when n is congruent to 2 mod 7.
A134160 is the case when n is congruent to 4 mod 7.
A134161 is the case when n is congruent to 5 mod 7.
A133180 is the union of this sequence, A134159, A134160, and A134161.

Links

Crossrefs

Cf. A000330, A000540, A119617, A134153, A134154, A133180, A134155, A134159, A134160, A134161.

Programs

  • Mathematica
    Table[(3(7n + 1)^4 + 6(7n + 1)^3 - 3 (7n + 1) + 1)/7, {n, 0, 100}] (* or *) Table[Sum[k^6, {k, 1, 7n + 1}]/Sum[k^2, {k, 1, 7n + 1}], {n, 0, 100}] (* Artur Jasinski *)
  • PARI
    Vec((1 + 2186*x + 13638*x^2 + 8498*x^3 + 373*x^4) / (1 - x)^5 + O(x^30)) \\ Colin Barker, Aug 12 2017

Formula

a(n) = (3(7n + 1)^4 + 6(7n + 1)^3 - 3 (7n + 1) + 1)/7.
a(n) = (Sum_{k=1..7n+1} k^6) / (Sum_{k=1..7n+1} k^2).
G.f.: -(1 + 2186*x + 13638*x^2 + 8498*x^3 + 373*x^4)/(-1+x)^5. - R. J. Mathar, Nov 14 2007
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n>4. - Colin Barker, Aug 12 2017

A134159 a(n) = 13 + 165*n + 756*n^2 + 1470*n^3 + 1029*n^4.

Original entry on oeis.org

13, 3433, 31591, 130351, 370273, 846613, 1679323, 3013051, 5017141, 7885633, 11837263, 17115463, 23988361, 32748781, 43714243, 57226963, 73653853, 93386521, 116841271, 144459103, 176705713, 214071493, 257071531, 306245611
Offset: 0

Views

Author

Artur Jasinski, Oct 10 2007

Keywords

Comments

A000540(n) is divisible by A000330(n) if and only if n is congruent to {1,2,4,5} mod 7 (see A047380). A134158 is the case when n is congruent to 1 mod 7. A134159 is the case when n is congruent to 2 mod 7. A134160 is the case when n is congruent to 4 mod 7. A134161 is the case when n is congruent to 5 mod 7. A133180 is the union of A134158 and A134159 and A134160 and A134161.

Crossrefs

Cf. A000330, A000540, A119617, A134153, A134154, A133180, A134158, A134160, A134161.

Programs

  • Mathematica
    Table[(3(7n + 2)^4 + 6(7n + 2)^3 - 3 (7n + 2) + 1)/7, {n, 0, 100}]
Table[Sum[k^6, {k, 1, 7n + 2}]/Sum[k^2, {k, 1, 7n + 2}], {n, 0, 100}] (* Artur Jasinski *)

Formula

a(n) = (3*(7*n + 2)^4 + 6*(7*n + 2)^3 - 3*(7*n + 2) + 1)/7.
a(n) = (Sum_{k=1..7n+2} k^6) / (Sum_{k=1..7n+2} k^2).
G.f.: -(13+3368*x+14556*x^2+6596*x^3+163*x^4)/(-1+x)^5. - R. J. Mathar, Nov 14 2007

A134160 a(n) = 163 + 1053*n + 2520*n^2 + 2646*n^3 + 1029*n^4.

Original entry on oeis.org

163, 7411, 49981, 180793, 477463, 1042303, 2002321, 3509221, 5739403, 8893963, 13198693, 18904081, 26285311, 35642263, 47299513, 61606333, 78936691, 99689251, 124287373, 153179113, 186837223, 225759151, 270467041, 321507733
Offset: 0

Views

Author

Artur Jasinski, Oct 10 2007

Keywords

Comments

A000540(n) is divisible by A000330(n) if and only n is congruent to {1,2,4,5} mod 7 (see A047380) A134158 is case when n is congruent to 1 mod 7 A134159 is case when n is congruent to 2 mod 7 A134160 is case when n is congruent to 4 mod 7 A134161 is case when n is congruent to 5 mod 7 A133180 is union of A134158 and A134159 and A134160 and A134161

Links

Crossrefs

Cf. A000330, A000540, A119617, A134153, A134154, A133180, A134158, A134159, A134161.

Programs

  • Mathematica
    Table[(3(7n + 4)^4 + 6(7n + 4)^3 - 3 (7n + 4) + 1)/7, {n, 0, 100}] (*Artur Jasinski*)
Table[Sum[k^6, {k, 1, 7n + 4}]/Sum[k^2, {k, 1, 7n + 4}], {n, 0, 100}] (*Artur Jasinski*)
LinearRecurrence[{5,-10,10,-5,1},{163,7411,49981,180793,477463},30] (* Harvey P. Dale, Jul 20 2024 *)
  • PARI
    a(n)=163+1053*n+2520*n^2+2646*n^3+1029*n^4 \\ Charles R Greathouse IV, Oct 07 2015

Formula

a(n) = (3*(7*n + 4)^4 + 6*(7*n + 4)^3 - 3*(7*n + 4) + 1)/7.
a(n) = sum(k=1..7*n+4, k^6) / sum(k=1..7*n+4, k^2).
G.f.: (163+6596*x+14556*x^2+3368*x^3+13*x^4)/(1-x)^5. - Colin Barker, May 25 2012

A134161 a(n) = 373 + 1947*n + 3780*n^2 + 3234*n^3 + 1029*n^4.

Original entry on oeis.org

373, 10363, 61723, 210901, 539041, 1151983, 2180263, 3779113, 6128461, 9432931, 13921843, 19849213, 27493753, 37158871, 49172671, 63887953, 81682213, 102957643, 128141131, 157684261, 192063313, 231779263, 277357783, 329349241
Offset: 0

Views

Author

Artur Jasinski, Oct 10 2007

Keywords

Comments

A000540(n) is divisible by A000330(n) if and only n is congruent to {1,2,4,5} mod 7 (see A047380) A134158 is case when n is congruent to 1 mod 7 A134159 is case when n is congruent to 2 mod 7 A134160 is case when n is congruent to 4 mod 7 A134161 is case when n is congruent to 5 mod 7 A133180 is union of A134158 and A134159 and A134160 and A134161.

Links

Crossrefs

Cf. A000330, A000540, A119617, A134153, A134154, A133180, A134158, A134159, A134160.

Programs

  • Mathematica
    Table[(3(7n + 5)^4 + 6(7n + 5)^3 - 3 (7n + 5) + 1)/7, {n, 0, 100}]
Table[Sum[k^6, {k, 1, 7n + 5}]/Sum[k^2, {k, 1, 7n + 5}], {n, 0, 100}]
LinearRecurrence[{5,-10,10,-5,1},{373,10363,61723,210901,539041},100] (* Harvey P. Dale, Nov 25 2012 *)
  • PARI
    a(n)=373+1947*n+3780*n^2+3234*n^3+1029*n^4 \\ Charles R Greathouse IV, Oct 07 2015

Formula

a(n) = (3*(7*n + 5)^4 + 6*(7*n + 5)^3 - 3*(7*n + 5) + 1)/7.
a(n) = (Sum_{k=1..7*n+5} k^6) / (Sum_{k=1..7*n+5} k^2).
G.f.: -(373+8498*x+13638*x^2+2186*x^3+x^4)/(-1+x)^5. - R. J. Mathar, Nov 14 2007
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) with a(0)=373, a(1)=10363, a(2)=61723, a(3)=210901, and a(4)=539041. - Harvey P. Dale, Nov 25 2012
Showing 1-5 of 5 results.

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