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.

A234094 Array {w(n,h)}: row n shows the weights, as defined in Comments, of the partitions of n, arranged in Mathematica order.

Original entry on oeis.org

1, 2, 3, 3, 5, 6, 4, 7, 6, 9, 10, 5, 9, 8, 12, 11, 14, 15, 6, 11, 10, 15, 9, 14, 18, 12, 17, 20, 21, 7, 13, 12, 18, 11, 17, 22, 16, 15, 21, 25, 19, 24, 27, 28, 8, 15, 14, 21, 13, 20, 26, 12, 19, 18, 25, 30, 17, 24, 23, 29, 33, 20, 27, 32, 35, 36, 9, 17, 16
Offset: 1

Views

Author

Clark Kimberling, Jan 01 2014

Keywords

Comments

The weight of a partition P = x(1)+x(2)+...+x(k) of n is introduced here as k*x(1)+(k-1)*x(2)+...+x(k), which is the number of steps needed to make P from the sum 1+1+...+1 = n by moving dividers (or parentheses) into the sum; see the Example section.

Examples

			Represent 1+1+1+1+1 as _1_1_1_1_1_.  The partition 2+2+1 matches the placement of dividers d indicated by _1_1d1_1d1d.  To place the 1st d takes 2 steps (starting at the 1st _); to place the 2nd d takes 2+2 = 4 steps (starting at the 1st _ ); to place the 3rd d takes 2+2+1 = 5 steps.  The total number of steps is 2+4+5 = 11, which is the 5th number in row 5 because 2+2+1 is the 5th partition of 5 in Mathematica ordering.  The first 6 rows are:
1
2 ... 3
3 ... 5 ... 6
4 ... 7 ... 6 ... 9 ... 10
5 ... 9 ... 8 ... 12 .. 11 .. 14 ... 15
6 ... 11 .. 10 .. 15 .. 9 ... 14 ... 18 .. 12 .. 17 .. 20 .. 21

Crossrefs

Cf. A234097, A234922, A234924.

Programs

  • Mathematica
    p[n_] := p[n] = IntegerPartitions[n]; q[n_] := q[n] = Length[p[n]]; v[n_] := v[n] = Table[n + 1 - i, {i, 1, n}]; w[n_, h_] := w[n, h] = Dot[p[n][[h]], v[Length[p[n][[h]]]]];
Flatten[Table[w[n, h], {n, 1, 9}, {h, 1, q[n]}]] (* A234094 *)
TableForm[Table[w[n, h], {n, 1, 9}, {h, 1, q[n]}]]

Formula

w(n,h) = dot product of (partition # h of n) and (k, k-1, ..., 1), where k = length of (partition # h of n).

A234105 Integers of the form (p*q*r*s - 1)/2, where p, q, r, s are distinct primes.

Original entry on oeis.org

577, 682, 892, 997, 1072, 1207, 1402, 1501, 1522, 1567, 1627, 1657, 1852, 1897, 1942, 1963, 2152, 2194, 2242, 2257, 2320, 2392, 2422, 2467, 2502, 2557, 2593, 2656, 2782, 2827, 2932, 3022, 3052, 3097, 3139, 3202, 3272, 3277, 3349, 3382, 3391, 3517, 3547, 3580
Offset: 1

Views

Author

Clark Kimberling, Jan 01 2014

Keywords

Crossrefs

Cf. A234498, A234499, A234500, A234093, A234097.

Programs

  • Mathematica
    t = Select[Range[1, 20000, 2], Map[Last, FactorInteger[#]] == Table[1, {4}] &]; Take[(t - 1)/2, 120] (* A234105 *)
v = Flatten[Position[PrimeQ[(t - 1)/2], True]] ; w = Table[t[[v[[n]]]], {n, 1, Length[v]}]  (* A234498 *)
(w - 1)/2 (* A234499 *) (* Peter J. C. Moses, Dec 23 2013 *)

Formula

-1 + A234500.

A234498 Products p*q*r*s of distinct primes for which (p*q*r*s - 1)/2 is prime.

Original entry on oeis.org

1155, 1995, 3135, 3255, 3315, 4935, 5115, 5187, 6783, 7035, 7095, 7215, 7395, 7455, 7755, 8463, 8547, 8715, 9867, 10335, 10455, 10695, 10815, 11055, 11715, 11739, 12243, 12903, 14595, 14835, 15855, 16107, 16359, 16779, 16863, 17043, 17255, 17355, 18183
Offset: 1

Views

Author

Clark Kimberling, Jan 01 2014

Keywords

Examples

			1155 = 3*5*7*11 is the least product of 4 distinct primes p,q,r,s for which (p*q*r*s-1)/2 is a prime: 577.

Crossrefs

Cf. A234105, A234499, A234500, A234097.

Programs

  • Mathematica
    t = Select[Range[1, 20000, 2], Map[Last, FactorInteger[#]] == Table[1, {4}] &]; Take[(t - 1)/2, 120] (* A234105 *)
v = Flatten[Position[PrimeQ[(t - 1)/2], True]] ; w = Table[t[[v[[n]]]], {n, 1, Length[v]}]  (* A234498 *)
(w - 1)/2 (* A234499 *) (* Peter J. C. Moses, Dec 23 2013 *)

A234924 Combined weight, as defined at A244094, of the distinct-parts partitions of n.

Original entry on oeis.org

1, 2, 8, 11, 22, 41, 60, 89, 136, 208, 275, 397, 526, 724, 978, 1279, 1646, 2172, 2752, 3518, 4492, 5620, 7010, 8742, 10809, 13280, 16346, 19937, 24200, 29373, 35436, 42548, 51153, 61039, 72794, 86632, 102615, 121268, 143209, 168458, 197753, 231833, 270983
Offset: 1

Views

Author

Clark Kimberling, Jan 01 2014

Keywords

Comments

These are the row sums of the array at A234923.

Crossrefs

Cf. A244093, A234097.

Programs

  • Mathematica
    z = 45; p[n_] := p[n] = Select[IntegerPartitions[n], Max[Length /@ Split@#] == 1 &]; q[n_] := q[n] = Length[p[n]]; v[n_] := v[n] = Table[n + 1 - i, {i, 1, n}]; w[n_, h_] := w[n, h] = Dot[p[n][[h]], v[Length[p[n][[h]]]]]; a[n_] := Sum[w[n, h], {h, 1, q[n]}]; Table[a[n], {n, 1, z}]

A234922 Array w(n,h), in which row n shows the weights (defined in Comments) of the partitions of n, arranged in reverse Mathematica order.

Original entry on oeis.org

1, 3, 2, 6, 5, 3, 10, 9, 6, 7, 4, 15, 14, 11, 12, 8, 9, 5, 21, 20, 17, 12, 18, 14, 9, 15, 10, 11, 6, 28, 27, 24, 19, 25, 21, 15, 16, 22, 17, 11, 18, 12, 13, 7, 36, 35, 32, 27, 20, 33, 29, 23, 24, 17, 30, 25, 18, 19, 12, 26, 20, 13, 21, 14, 15, 8, 45, 44, 41
Offset: 1

Views

Author

Clark Kimberling, Jan 01 2014

Keywords

Comments

The weight of a partition P = x(1)+ x(2)+...+x(k) of n is introduced at A234094 as k*x(1) + (k-1)*x(2) + ... + x(k).

Examples

			Represent 1+1+1+1+1 as _1_1_1_1_1_.  The partition 1+2+2 matches the placement of dividers d indicated by _1d1_1d1_1d. To place the 1st d takes 1 step (starting at the 1st '_'); to place the 2nd d takes 1+2 steps (starting at the 1st '_'); to place the 3rd d takes 1+2+2 steps. The total number of steps is 2+3+5 = 9, the 3rd number in row 5, because 1+2+2 is the 3rd partition of 5 in reverse Mathematica ordering. The first 6 rows:
1
3    2
6    5    3
10   9    6    7    4
15   14   11   12   8    9    5
21   20   17   12   18   14   9   15   10   11   6

Crossrefs

Cf. A234094, A234097.

Programs

  • Mathematica
    p[n_] := p[n] = Reverse[IntegerPartitions[n]]; q[n_] := q[n] = Length[p[n]]; v[n_] := v[n] = Table[n + 1 - i, {i, 1, n}]; w[n_, h_] := w[n, h] = Dot[p[n][[h]], v[Length[p[n][[h]]]]];
Flatten[Table[w[n, h], {n, 1, 9}, {h, 1, q[n]}]] (* A234094 *)
TableForm[Table[w[n, h], {n, 1, 9}, {h, 1, q[n]}]]

Formula

w(n,h) = dot product of (partition # h of n) and (k, k-1, ..., 1), where k = length of (partition # h of n).
Showing 1-5 of 5 results.

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