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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A195827 Triangle read by rows with T(n,k) = n - A085787(k), n>=1, k>=1, if (n - A085787(k))>=0.

Original entry on oeis.org

0, 1, 2, 3, 0, 4, 1, 5, 2, 6, 3, 0, 7, 4, 1, 8, 5, 2, 9, 6, 3, 10, 7, 4, 11, 8, 5, 12, 9, 6, 0, 13, 10, 7, 1, 14, 11, 8, 2, 15, 12, 9, 3, 16, 13, 10, 4, 17, 14, 11, 5, 0, 18, 15, 12, 6, 1, 19, 16, 13, 7, 2, 20, 17, 14, 8, 3, 21, 18, 15, 9, 4, 22, 19, 16, 10, 5
Offset: 1

Views

Author

Omar E. Pol, Sep 24 2011

Keywords

Comments

Also triangle read by rows in which column k lists the nonnegative integers A001477 starting at the row A085787(k).
This sequence is related to the generalized heptagonal numbers A085787, A195837 and A036820 in the same way as A195310 is related to the generalized pentagonal numbers A001318, A175003 and A000041. See comments in A195825.

Examples

			Written as a triangle:
.  0;
.  1;
.  2;
.  3,  0;
.  4,  1;
.  5,  2;
.  6,  3,  0;
.  7,  4,  1;
.  8,  5,  2;
.  9,  6,  3;
. 10,  7,  4;
. 11,  8,  5;
. 12,  9,  6,  0;
. 13, 10,  7,  1;
. 14, 11,  8,  2;

Crossrefs

Cf. A036820, A085787, A195310, A195825, A195826, A195828, A195829, A195830, A195831, A195832, A195833, A195837.

A238442 Triangle read by rows demonstrating Euler's pentagonal theorem for the sum of divisors.

Original entry on oeis.org

1, 1, 2, 3, 1, 4, 3, 7, 4, -5, 6, 7, -1, 12, 6, -3, -7, 8, 12, -4, -1, 15, 8, -7, -3, 13, 15, -6, -4, 18, 13, -12, -7, 12, 18, -8, -6, 12, 28, 12, -15, -12, 1, 14, 28, -13, -8, 3, 24, 14, -18, -15, 4, 15, 24, 24, -12, -13, 7, 1, 31, 24, -28, -18, 6, 3
Offset: 1

Views

Author

Omar E. Pol, Feb 26 2014

Keywords

Comments

The law found by Leonhard Euler for the sum of divisors of n is that S(n) = S(n - 1) + S(n - 2) - S(n - 5) - S(n - 7) + S(n - 12) + S(n - 15) - S(n - 22) - S(n - 26) + S(n - 35) + S(n - 40) + ..., where the constants are the positive generalized pentagonal numbers, and S(0) = n, which is also a positive member of A001318.
Therefore column k lists A001318(k) together with the elements of A000203, starting at row A001318(k), but with all elements of column k multiplied by A057077(k-1).
The first element of column k is A057077(k-1)*A001318(k)which is also the last term of row A001318(k).
For Euler's pentagonal theorem for the partition numbers see A175003.
Note that both of Euler's pentagonal theorems refer to generalized pentagonal numbers (A001318), not to pentagonal numbers (A000326).

Examples

			Triangle begins:
   1;
   1,   2;
   3,   1;
   4,   3;
   7,   4,  -5;
   6,   7,  -1;
  12,   6,  -3,  -7;
   8,  12,  -4,  -1;
  15,   8,  -7,  -3;
  13,  15,  -6,  -4;
  18,  13, -12,  -7;
  12,  18,  -8,  -6,  12;
  28,  12, -15, -12,   1;
  14,  28, -13,  -8,   3;
  24,  14, -18, -15,   4,  15;
  24,  24, -12, -13,   7,   1;
  31,  24, -28, -18,   6,   3;
  18,  31, -14, -12,  12,   4;
  39,  18, -24, -28,   8,   7;
  20,  39, -24, -14,  15,   6;
  42,  20, -31, -24,  13,  12;
  32,  42, -18, -24,  18,   8, -22;
  36,  32, -39, -31,  12,  15,  -1;
  24,  36, -20, -18,  28,  13,  -3;
  60,  24, -42, -39,  14,  18,  -4;
  31,  60, -32, -20,  24,  12,  -7, -26;
  ...
For n = 21 the sum of divisors of 21 is 1 + 3 + 7 + 21 = 32. On the other hand, from Euler's Pentagonal Number Theorem we have that the sum of divisors of 21 is S_21 = S_20 + S_19 - S_16 - S_14 + S_9 + S_6, the same as the sum of the 21st row of triangle: 42 + 20 - 31 - 24 + 13 + 12 = 32, equaling the sum of divisors of 21.
For n = 22 the sum of divisors of 22 is 1 + 2 + 11 + 22 = 36. On the other hand, from Euler's Pentagonal Number Theorem we have that the sum of divisors of 22 is S_22 = S_21 + S_20 - S_17 - S_15 + S_10 + S_7 - S_0, the same as the sum of the 22nd row of triangle is 32 + 42 - 18 - 24 + 18 + 8 - 22 = 36, equaling the sum of divisors of 22. Note that S_0 = n, hence in this case S_0 = 22.

Links

Crossrefs

Row sums give A000203, the sum of divisors of n.
Row n has length A235963(n).
Cf. A001318, A027750, A057077, A175003, A196020, A237270, A237273.

Programs

  • Mathematica
    rows = m = 18;
a057077[n_] := {1, 1, -1, -1}[[Mod[n, 4] + 1]];
a001318[n_] := (1/8)((2n + 1) Mod[n, 2] + 3n^2 + 2n);
a235963[n_] := Flatten[Table[k, {k, 0, m}, {(k+1)/(Mod[k, 2]+1)}]][[n+1]];
T[n_, k_] := If[n == a001318[k] && k == a235963[n], a001318[k] a057077[k - 1], a057077[k - 1] DivisorSigma[1, n - a001318[k]]];
Table[T[n, k], {n, 1, m}, {k, 1, a235963[n]}] // Flatten (* Jean-François Alcover, Nov 29 2018 *)

Formula

T(n,k) = A057077(k-1)*A001318(k), if n = A001318(k) and k = A235963(n). Otherwise T(n,k) = A057077(k-1)*A000203(n - A001318(k)), n >= 1, 1 <= k <= A235963(n).

A210944 Triangle read by rows with T(n,k) = n - A195818(k), n>=1, k>=1, if (n - A195818(k))>=0.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 0, 11, 1, 12, 2, 13, 3, 0, 14, 4, 1, 15, 5, 2, 16, 6, 3, 17, 7, 4, 18, 8, 5, 19, 9, 6, 20, 10, 7, 21, 11, 8, 22, 12, 9, 23, 13, 10, 24, 14, 11, 25, 15, 12, 26, 16, 13, 27, 17, 14, 28, 18, 15, 29, 19, 16, 30, 20, 17, 31, 21, 18
Offset: 1

Views

Author

Omar E. Pol, Jun 16 2012

Keywords

Comments

Also triangle read by rows in which column k lists the nonnegative integers A001477 starting at the row A195818(k).
This sequence is related to the generalized 14-gonal numbers A195818, A210954 and A210964 in the same way as A195310 is related to the generalized pentagonal numbers A001318, A175003 and A000041. See comments in A195825.

Examples

			Written as an irregular triangle:
0;
1;
2;
3;
4;
5;
6;
7;
8;
9;
10, 0;
11, 1;
12, 2;
13, 3,  0;
14, 4,  1;
15, 5,  2;
16, 6,  3;
17, 7,  4;
18, 8,  5;
19, 9,  6;

Crossrefs

Cf. A195310, A195825, A195826, A195827, A195828, A195829, A195830, A195831, A195832, A195833, A195818, A210954, A210964.

A244964 Number of distinct generalized pentagonal numbers dividing n.

Original entry on oeis.org

1, 2, 1, 2, 2, 2, 2, 2, 1, 3, 1, 3, 1, 3, 3, 2, 1, 2, 1, 3, 2, 3, 1, 3, 2, 3, 1, 3, 1, 4, 1, 2, 1, 2, 4, 3, 1, 2, 1, 4, 1, 3, 1, 3, 3, 2, 1, 3, 2, 3, 2, 3, 1, 2, 2, 3, 2, 2, 1, 5, 1, 2, 2, 2, 2, 3, 1, 2, 1, 6, 1, 3, 1, 2, 3, 2, 3, 3, 1, 4, 1, 2, 1, 4, 2, 2, 1, 3, 1, 4, 2, 3, 1, 2, 2, 3, 1, 3, 1, 4, 1, 3, 1, 3, 5
Offset: 1

Views

Author

Omar E. Pol, Jul 10 2014

Keywords

Comments

For more information about the generalized pentagonal numbers see A001318.

Examples

			For n = 10 the generalized pentagonal numbers <= 10 are [0, 1, 2, 5, 7]. There are three generalized pentagonal numbers that divide 10; they are [1, 2, 5], so a(10) = 3.

Links

Crossrefs

Cf. A000005, A001221, A001318, A001511, A005086, A006519, A007862, A027750, A046951, A080995, A147645, A175003, A236103, A238442, A239930.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, 1 &, IntegerQ[Sqrt[24*# + 1]] &]; Array[a, 100] (* Amiram Eldar, Dec 31 2023 *)
  • PARI
    a(n) = sumdiv(n, d, issquare(24*d + 1)); \\ Amiram Eldar, Dec 31 2023

Formula

From Amiram Eldar, Dec 31 2023: (Start)
a(n) = Sum_{d|n} A080995(d).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 6 - 2*Pi/sqrt(3) = 2.372401... . (End)

A244962 Generalized pentagonal numbers that are also partition numbers.

Original entry on oeis.org

1, 2, 5, 7, 15, 22, 77, 176, 4565
Offset: 1

Views

Author

Omar E. Pol, Jul 10 2014

Keywords

Comments

Intersection of A001318 and A000041.
No more terms < A000041(2*10^6), a number with 1569 digits. - Lars Blomberg, Jul 15 2014

Crossrefs

Cf. A000041, A001318, A175003.

Programs

  • Mathematica
    nmax=PartitionsP[50]; Intersection[Table[PartitionsP[n],{n,0,50}],Table[n*(n+1)/6,{n,Select[Range[0,nmax],Mod[#,3]!=1&]}]] (* Vaclav Kotesovec, Jul 15 2014 *)
Previous Showing 21-25 of 25 results.

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