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.

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A034666 Sum of n-th powers of divisors of 36.

Original entry on oeis.org

9, 91, 1911, 55261, 1813539, 62672701, 2214363531, 79017297541, 2832604648419, 101763873519661, 3659794373013051, 131686747008947221, 4739547370427122899, 170602660692644492221, 6141318334706291980971
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A018256 (divisors of 36).

Programs

Formula

G.f.: -(25474176*x^8 -49533120*x^7 +36978120*x^6 -13929552*x^5 +2901990*x^4 -342390*x^3 +22295*x^2 -728*x +9) / ((x -1)*(2*x -1)*(3*x -1)*(4*x -1)*(6*x -1)*(9*x -1)*(12*x -1)*(18*x -1)*(36*x -1)). - Colin Barker, Apr 20 2014
From Wesley Ivan Hurt, Aug 04 2025: (Start)
a(n) = (1 + 2^n + 4^n) * (1 + 3^n + 9^n).
a(n) = 91*a(n-1) - 3185*a(n-2) + 57065*a(n-3) - 580398*a(n-4) + 3482388*a(n-5) - 12326040*a(n-6) + 24766560*a(n-7) - 25474176*a(n-8) + 10077696*a(n-9). (End)

A354235 Heinz numbers of integer partitions with at least one part divisible by 3.

Original entry on oeis.org

5, 10, 13, 15, 20, 23, 25, 26, 30, 35, 37, 39, 40, 45, 46, 47, 50, 52, 55, 60, 61, 65, 69, 70, 73, 74, 75, 78, 80, 85, 89, 90, 91, 92, 94, 95, 100, 103, 104, 105, 110, 111, 113, 115, 117, 120, 122, 125, 130, 135, 137, 138, 140, 141, 143, 145, 146, 148, 150
Offset: 1

Views

Author

Gus Wiseman, May 23 2022

Keywords

Comments

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

Examples

			The terms together with their prime indices begin:
    5: {3}
   10: {1,3}
   13: {6}
   15: {2,3}
   20: {1,1,3}
   23: {9}
   25: {3,3}
   26: {1,6}
   30: {1,2,3}
   35: {3,4}
   37: {12}
   39: {2,6}
   40: {1,1,1,3}
   45: {2,2,3}
   46: {1,9}
   47: {15}
   50: {1,3,3}
   52: {1,1,6}
   55: {3,5}
   60: {1,1,2,3}

Crossrefs

For 4 instead of 3 we have A046101, counted by A295342.
This sequence ranks the partitions counted by A295341, compositions A335464.
For 2 instead of 3 we have A324929 (and A013929), counted by A047967.
A001222 counts prime factors with multiplicity, distinct A001221.
A004250 counts partitions with some part > 2, compositions A008466.
A004709 lists numbers divisible by no cube, counted by A000726.
A036966 lists 3-full numbers, counted by A100405.
A046099 lists non-cubefree numbers.
A056239 adds up prime indices, row sums of A112798 and A296150.
A124010 gives prime signature, sorted A118914.
A354234 counts partitions of n with at least one part divisible by k.
Cf. A000720, A003963, A008483, A018256, A062739, A064428, A117485, A181819, A339222, A353508.

Programs

  • Mathematica
    Select[Range[100],MemberQ[PrimePi/@First/@If[#==1,{},FactorInteger[#]]/3,_?IntegerQ]&]

A018259 Divisors of 44.

Original entry on oeis.org

1, 2, 4, 11, 22, 44
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A018253, A018256, A018261.

Programs

A018338 Divisors of 216.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 27, 36, 54, 72, 108, 216
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

There are 170 ways to partition 216 into its distinct divisors (see A033630). This would be a record, if it weren't for 120 (with 279 ways), 168 (with 198 ways) and 180 (with 752 ways). - Alonso del Arte, Sep 24 2017
The biquadratefree 3-smooth numbers: intersection of A046100 and A003586. - Amiram Eldar, Sep 26 2023

Links

Crossrefs

Cf. A003586, A018253, A018256, A018264, A018287, A046100.

Programs

Previous Showing 21-24 of 24 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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