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 1-3 of 3 results.

A294145 Sum of the smaller parts of the partitions of n into two parts with larger part squarefree.

Original entry on oeis.org

0, 1, 1, 3, 2, 4, 3, 6, 9, 12, 10, 14, 11, 15, 12, 17, 22, 28, 34, 41, 37, 44, 40, 48, 56, 64, 59, 67, 60, 68, 61, 70, 79, 89, 82, 93, 104, 116, 109, 122, 135, 149, 142, 157, 149, 163, 154, 169, 184, 199, 214, 230, 219, 235, 251, 268, 285, 303, 292, 311, 299
Offset: 1

Views

Author

Wesley Ivan Hurt, Oct 23 2017

Keywords

Links

Crossrefs

Cf. A008683, A008966, A294146.

Programs

  • Mathematica
    Table[Sum[i*MoebiusMu[n - i]^2, {i, Floor[n/2]}], {n, 80}]
Table[Total[Select[IntegerPartitions[n,{2}],SquareFreeQ[#[[1]]]&][[All,2]]],{n,70}] (* Harvey P. Dale, Dec 26 2020 *)

Formula

a(n) = Sum_{i=1..floor(n/2)} i * mu(n-i)^2, where mu is the Möbius function (A008683).

A294263 Sum of the larger parts of the partitions of n into two parts with smaller part squarefree > 1.

Original entry on oeis.org

0, 0, 0, 2, 3, 7, 9, 11, 13, 20, 23, 32, 36, 47, 52, 57, 62, 67, 72, 87, 93, 110, 117, 124, 131, 151, 159, 181, 190, 214, 224, 234, 244, 271, 282, 293, 304, 334, 346, 358, 370, 403, 416, 451, 465, 502, 517, 532, 547, 562, 577, 618, 634, 650, 666, 682, 698
Offset: 1

Views

Author

Wesley Ivan Hurt, Oct 26 2017

Keywords

Links

Crossrefs

Cf. A008683, A008966, A294146.

Programs

  • Mathematica
    Table[Sum[(n - i) MoebiusMu[i]^2, {i, 2, Floor[n/2]}], {n, 80}]

Formula

a(n) = Sum_{i=2..floor(n/2)} (n - i) * mu(i)^2, where mu is the Möbius function (A008683).

A294283 Sum of the larger parts of the partitions of n into two distinct parts with smaller part squarefree.

Original entry on oeis.org

0, 0, 2, 3, 7, 9, 15, 18, 21, 24, 33, 37, 48, 53, 66, 72, 78, 84, 90, 96, 113, 120, 139, 147, 155, 163, 185, 194, 218, 228, 254, 265, 276, 287, 316, 328, 340, 352, 384, 397, 410, 423, 458, 472, 509, 524, 563, 579, 595, 611, 627, 643, 686, 703, 720, 737, 754
Offset: 1

Views

Author

Wesley Ivan Hurt, Oct 26 2017

Keywords

Comments

Sum of the lengths of the distinct rectangles with squarefree width and positive integer length such that L + W = n, W < L. For example, a(14) = 53; the rectangles are 1 X 13, 2 X 12, 3 X 11, 5 X 9, 6 X 8. The sum of the lengths is then 13 + 12 + 11 + 9 + 8 = 53. - Wesley Ivan Hurt, Nov 12 2017

Examples

			a(5) = 7; the partitions of 5 into two distinct parts are (4,1) and (3,2). The smaller parts are both squarefree, so the sum of the larger parts is 4+3 = 7.
a(10) = 24; the partitions of 10 into two distinct parts are (9,1), (8,2), (7,3) and (6,4). Of the smaller parts, only 1, 2, and 3 are squarefree, so we add the larger parts of those partitions to get 9+8+7 = 24.

Links

Crossrefs

Cf. A008683, A008966, A262869, A262871, A294146.

Programs

  • Mathematica
    Table[Sum[(n - i) MoebiusMu[i]^2, {i, Floor[(n-1)/2]}], {n, 60}]
  • PARI
    a(n) = sum(i=1, (n-1)\2, (n-i)*moebius(i)^2); \\ Michel Marcus, Nov 08 2017

Formula

a(n) = Sum_{i=1..floor((n-1)/2)} (n - i) * mu(i)^2, where mu is the Möbius function (A008683).
Showing 1-3 of 3 results.

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

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