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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A291511 The arithmetic function uhat(n,2,6).

Original entry on oeis.org

-23, -24, -23, -24, -23, -24, -23, -24, -27, -30, -33, -36, -39, -42, -45, -48, -51, -54, -57, -60, -63, -66, -69, -72, -75, -78, -81, -84, -87, -90, -93, -96, -99, -102, -105, -108, -111, -114, -117, -120, -123, -126, -129, -132, -135, -138, -141, -144, -147, -150, -153, -156, -159, -162, -165, -168, -171, -174, -177, -180, -183, -186, -189, -192, -195, -198, -201, -204, -207, -210
Offset: 1

Views

Author

Robert Price, Aug 25 2017

Keywords

Links

Crossrefs

Cf. A289435, A289436, A289437, A289438, A289439, A289440, A289441.

Programs

  • Mathematica
    delta[r_, k_, d_] := If[r < k, (k - r)*r - (d - 1), If[k < r && r < d, (d - r)*(r - k) - (d - 1), If[k == r && r == d, d - 1, 0]]] uhat[n_, m_, h_] := (dx = Divisors[n]; dmin = n; For[i = 1, i ≤ Length[dx], i++, d = dx[[i]]; k = m - d*Ceiling[m/d] + d; r = h - d*Ceiling[h/d] + d; If[h ≤ Min[k, d - 1], dmin = Min[dmin, n, (h*Ceiling[m/d] - h + 1)*d, h*m - h*h + 1], dmin = Min[dmin, n, h*m - h*h + 1 - delta[r, k, d]]]]; dmin) Table[uhat[n, 2, 6], {n, 1, 70}]

A291512 The arithmetic function uhat(n,2,7).

Original entry on oeis.org

-34, -34, -34, -34, -34, -34, -34, -34, -36, -40, -44, -48, -52, -56, -60, -64, -68, -72, -76, -80, -84, -88, -92, -96, -100, -104, -108, -112, -116, -120, -124, -128, -132, -136, -140, -144, -148, -152, -156, -160, -164, -168, -172, -176, -180, -184, -188, -192, -196, -200, -204, -208, -212, -216, -220, -224, -228, -232, -236, -240, -244, -248, -252, -256, -260, -264, -268, -272, -276, -280
Offset: 1

Views

Author

Robert Price, Aug 25 2017

Keywords

Comments

Appears to decrement by 4 for each successive term after a(9) = -36. - Chai Wah Wu, Sep 12 2018

Links

Crossrefs

Cf. A289435, A289436, A289437, A289438, A289439, A289440, A289441.

Programs

  • Mathematica
    delta[r_, k_, d_] := If[r < k, (k - r)*r - (d - 1), If[k < r && r < d, (d - r)*(r - k) - (d - 1), If[k == r && r == d, d - 1, 0]]] uhat[n_, m_, h_] := (dx = Divisors[n]; dmin = n; For[i = 1, i ≤ Length[dx], i++, d = dx[[i]]; k = m - d*Ceiling[m/d] + d; r = h - d*Ceiling[h/d] + d; If[h ≤ Min[k, d - 1], dmin = Min[dmin, n, (h*Ceiling[m/d] - h + 1)*d, h*m - h*h + 1], dmin = Min[dmin, n, h*m - h*h + 1 - delta[r, k, d]]]]; dmin) Table[uhat[n, 2, 7], {n, 1, 70}]

A291513 The arithmetic function uhat(n,2,8).

Original entry on oeis.org

-47, -48, -47, -48, -47, -48, -47, -48, -47, -50, -55, -60, -65, -70, -75, -80, -85, -90, -95, -100, -105, -110, -115, -120, -125, -130, -135, -140, -145, -150, -155, -160, -165, -170, -175, -180, -185, -190, -195, -200, -205, -210, -215, -220, -225, -230, -235, -240, -245, -250, -255, -260, -265, -270, -275, -280, -285, -290, -295, -300, -305, -310, -315, -320, -325, -330, -335, -340, -345, -350
Offset: 1

Views

Author

Robert Price, Aug 25 2017

Keywords

Links

Crossrefs

Cf. A289435, A289436, A289437, A289438, A289439, A289440, A289441.

Programs

  • Mathematica
    delta[r_, k_, d_] := If[r < k, (k - r)*r - (d - 1), If[k < r && r < d, (d - r)*(r - k) - (d - 1), If[k == r && r == d, d - 1, 0]]] uhat[n_, m_, h_] := (dx = Divisors[n]; dmin = n; For[i = 1, i ≤ Length[dx], i++, d = dx[[i]]; k = m - d*Ceiling[m/d] + d; r = h - d*Ceiling[h/d] + d; If[h ≤ Min[k, d - 1], dmin = Min[dmin, n, (h*Ceiling[m/d] - h + 1)*d, h*m - h*h + 1], dmin = Min[dmin, n, h*m - h*h + 1 - delta[r, k, d]]]]; dmin) Table[uhat[n, 2, 8], {n, 1, 70}]

A291515 The arithmetic function uhat(n,3,6).

Original entry on oeis.org

-17, -17, -19, -17, -17, -19, -17, -17, -19, -20, -22, -24, -26, -28, -30, -32, -34, -36, -38, -40, -42, -44, -46, -48, -50, -52, -54, -56, -58, -60, -62, -64, -66, -68, -70, -72, -74, -76, -78, -80, -82, -84, -86, -88, -90, -92, -94, -96, -98, -100, -102, -104, -106, -108, -110, -112, -114, -116, -118, -120, -122, -124, -126, -128, -130, -132, -134, -136, -138, -140
Offset: 1

Views

Author

Robert Price, Aug 25 2017

Keywords

Links

Crossrefs

Cf. A289435, A289436, A289437, A289438, A289439, A289440, A289441.

Programs

  • Mathematica
    delta[r_, k_, d_] := If[r < k, (k - r)*r - (d - 1), If[k < r && r < d, (d - r)*(r - k) - (d - 1), If[k == r && r == d, d - 1, 0]]] uhat[n_, m_, h_] := (dx = Divisors[n]; dmin = n; For[i = 1, i ≤ Length[dx], i++, d = dx[[i]]; k = m - d*Ceiling[m/d] + d; r = h - d*Ceiling[h/d] + d; If[h ≤ Min[k, d - 1], dmin = Min[dmin, n, (h*Ceiling[m/d] - h + 1)*d, h*m - h*h + 1], dmin = Min[dmin, n, h*m - h*h + 1 - delta[r, k, d]]]]; dmin) Table[uhat[n, 3, 6], {n, 1, 70}]

Formula

Conjectures from Chai Wah Wu, Jun 10 2025: (Start)
a(n) = 2*a(n-1) - a(n-2) for n > 11.
G.f.: x*(-x^10 + x^9 - 2*x^8 - 2*x^7 + 4*x^6 - 2*x^5 - 2*x^4 + 4*x^3 - 2*x^2 + 17*x - 17)/(x - 1)^2. (End)

A291516 The arithmetic function uhat(n,3,7), negated.

Original entry on oeis.org

27, 27, 27, 27, 27, 27, 27, 27, 27, 30, 33, 36, 39, 42, 45, 48, 51, 54, 57, 60, 63, 66, 69, 72, 75, 78, 81, 84, 87, 90, 93, 96, 99, 102, 105, 108, 111, 114, 117, 120, 123, 126, 129, 132, 135, 138, 141, 144, 147, 150, 153, 156, 159, 162, 165, 168, 171, 174, 177, 180, 183, 186, 189, 192, 195, 198, 201, 204, 207, 210
Offset: 1

Views

Author

Robert Price, Aug 25 2017

Keywords

Links

Crossrefs

Cf. A289435, A289436, A289437, A289438, A289439, A289440, A289441.

Programs

  • Mathematica
    delta[r_, k_, d_] := If[r < k, (k - r)*r - (d - 1), If[k < r && r < d, (d - r)*(r - k) - (d - 1), If[k == r && r == d, d - 1, 0]]]; uhat[n_, m_, h_] := (dx = Divisors[n]; dmin = n; For[i = 1, i ≤ Length[dx], i++, d = dx[[i]]; k = m - d*Ceiling[m/d] + d; r = h - d*Ceiling[h/d] + d; If[h ≤ Min[k, d - 1], dmin = Min[dmin, n, (h*Ceiling[m/d] - h + 1)*d, h*m - h*h + 1], dmin = Min[dmin, n, h*m - h*h + 1 - delta[r, k, d]]]]; dmin); Table[uhat[n, 3, 7], {n, 1, 70}]

A291517 The arithmetic function uhat(n,3,8).

Original entry on oeis.org

-39, -39, -39, -39, -39, -39, -39, -39, -39, -40, -44, -48, -52, -56, -60, -64, -68, -72, -76, -80, -84, -88, -92, -96, -100, -104, -108, -112, -116, -120, -124, -128, -132, -136, -140, -144, -148, -152, -156, -160, -164, -168, -172, -176, -180, -184, -188, -192, -196, -200, -204, -208, -212, -216, -220, -224, -228, -232, -236, -240, -244, -248, -252, -256, -260, -264, -268, -272, -276, -280
Offset: 1

Views

Author

Robert Price, Aug 25 2017

Keywords

Links

Crossrefs

Cf. A289435, A289436, A289437, A289438, A289439, A289440, A289441.

Programs

  • Mathematica
    delta[r_, k_, d_] := If[r < k, (k - r)*r - (d - 1), If[k < r && r < d, (d - r)*(r - k) - (d - 1), If[k == r && r == d, d - 1, 0]]] uhat[n_, m_, h_] := (dx = Divisors[n]; dmin = n; For[i = 1, i ≤ Length[dx], i++, d = dx[[i]]; k = m - d*Ceiling[m/d] + d; r = h - d*Ceiling[h/d] + d; If[h ≤ Min[k, d - 1], dmin = Min[dmin, n, (h*Ceiling[m/d] - h + 1)*d, h*m - h*h + 1], dmin = Min[dmin, n, h*m - h*h + 1 - delta[r, k, d]]]]; dmin) Table[uhat[n, 3, 8], {n, 1, 70}]

A291521 The arithmetic function uhat(n,4,6).

Original entry on oeis.org

-11, -12, -11, -12, -11, -12, -11, -12, -11, -12, -11, -12, -13, -14, -15, -16, -17, -18, -19, -20, -21, -22, -23, -24, -25, -26, -27, -28, -29, -30, -31, -32, -33, -34, -35, -36, -37, -38, -39, -40, -41, -42, -43, -44, -45, -46, -47, -48, -49, -50, -51, -52, -53, -54, -55, -56, -57, -58, -59, -60, -61, -62, -63, -64, -65, -66, -67, -68, -69, -70
Offset: 1

Views

Author

Robert Price, Aug 25 2017

Keywords

Links

Crossrefs

Cf. A289435, A289436, A289437, A289438, A289439, A289440, A289441.

Programs

  • Mathematica
    delta[r_, k_, d_] := If[r < k, (k - r)*r - (d - 1), If[k < r && r < d, (d - r)*(r - k) - (d - 1), If[k == r && r == d, d - 1, 0]]] uhat[n_, m_, h_] := (dx = Divisors[n]; dmin = n; For[i = 1, i ≤ Length[dx], i++, d = dx[[i]]; k = m - d*Ceiling[m/d] + d; r = h - d*Ceiling[h/d] + d; If[h ≤ Min[k, d - 1], dmin = Min[dmin, n, (h*Ceiling[m/d] - h + 1)*d, h*m - h*h + 1], dmin = Min[dmin, n, h*m - h*h + 1 - delta[r, k, d]]]]; dmin) Table[uhat[n, 4, 6], {n, 1, 70}]

A291522 The arithmetic function uhat(n,4,7).

Original entry on oeis.org

-20, -20, -20, -20, -20, -20, -20, -20, -20, -20, -22, -24, -26, -28, -30, -32, -34, -36, -38, -40, -42, -44, -46, -48, -50, -52, -54, -56, -58, -60, -62, -64, -66, -68, -70, -72, -74, -76, -78, -80, -82, -84, -86, -88, -90, -92, -94, -96, -98, -100, -102, -104, -106, -108, -110, -112, -114, -116, -118, -120, -122, -124, -126, -128, -130, -132, -134, -136, -138, -140
Offset: 1

Views

Author

Robert Price, Aug 25 2017

Keywords

Links

Crossrefs

Cf. A289435, A289436, A289437, A289438, A289439, A289440, A289441.

Programs

  • Mathematica
    delta[r_, k_, d_] := If[r < k, (k - r)*r - (d - 1), If[k < r && r < d, (d - r)*(r - k) - (d - 1), If[k == r && r == d, d - 1, 0]]] uhat[n_, m_, h_] := (dx = Divisors[n]; dmin = n; For[i = 1, i ≤ Length[dx], i++, d = dx[[i]]; k = m - d*Ceiling[m/d] + d; r = h - d*Ceiling[h/d] + d; If[h ≤ Min[k, d - 1], dmin = Min[dmin, n, (h*Ceiling[m/d] - h + 1)*d, h*m - h*h + 1], dmin = Min[dmin, n, h*m - h*h + 1 - delta[r, k, d]]]]; dmin) Table[uhat[n, 4, 7], {n, 1, 70}]

A291523 The arithmetic function uhat(n,4,8).

Original entry on oeis.org

-31, -32, -31, -34, -31, -32, -31, -34, -31, -32, -33, -36, -39, -42, -45, -48, -51, -54, -57, -60, -63, -66, -69, -72, -75, -78, -81, -84, -87, -90, -93, -96, -99, -102, -105, -108, -111, -114, -117, -120, -123, -126, -129, -132, -135, -138, -141, -144, -147, -150, -153, -156, -159, -162, -165, -168, -171, -174, -177, -180, -183, -186, -189, -192, -195, -198, -201, -204, -207, -210
Offset: 1

Views

Author

Robert Price, Aug 25 2017

Keywords

Links

Crossrefs

Cf. A289435, A289436, A289437, A289438, A289439, A289440, A289441.

Programs

  • Mathematica
    delta[r_, k_, d_] := If[r < k, (k - r)*r - (d - 1), If[k < r && r < d, (d - r)*(r - k) - (d - 1), If[k == r && r == d, d - 1, 0]]] uhat[n_, m_, h_] := (dx = Divisors[n]; dmin = n; For[i = 1, i ≤ Length[dx], i++, d = dx[[i]]; k = m - d*Ceiling[m/d] + d; r = h - d*Ceiling[h/d] + d; If[h ≤ Min[k, d - 1], dmin = Min[dmin, n, (h*Ceiling[m/d] - h + 1)*d, h*m - h*h + 1], dmin = Min[dmin, n, h*m - h*h + 1 - delta[r, k, d]]]]; dmin) Table[uhat[n, 4, 8], {n, 1, 70}]

A291567 The arithmetic function uhat(n,5,2).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 7, 7, 5, 7, 6, 7, 7, 5, 7, 7, 6, 7, 5, 7, 7, 7, 6, 5, 7, 7, 7, 7, 5, 7, 7, 7, 7, 5, 6, 7, 7, 7, 5, 7, 6, 7, 7, 5, 7, 7, 6, 7, 5, 7, 7, 7, 6, 5, 7, 7, 7, 7, 5, 7, 7, 7, 7, 5, 6, 7, 7, 7, 5
Offset: 1

Views

Author

Robert Price, Aug 26 2017

Keywords

Comments

The sequence appears to be equal to uhat(n,5,3), at least to 10000 terms.

Links

Crossrefs

Cf. A289435, A289436, A289437, A289438, A289439, A289440, A289441.

Programs

  • Mathematica
    delta[r_, k_, d_] := If[r < k, (k - r)*r - (d - 1), If[k < r && r < d, (d - r)*(r - k) - (d - 1), If[k == r && r == d, d - 1, 0]]] uhat[n_, m_, h_] := (dx = Divisors[n]; dmin = n; For[i = 1, i ≤ Length[dx], i++, d = dx[[i]]; k = m - d*Ceiling[m/d] + d; r = h - d*Ceiling[h/d] + d; If[h ≤ Min[k, d - 1], dmin = Min[dmin, n, (h*Ceiling[m/d] - h + 1)*d, h*m - h*h + 1], dmin = Min[dmin, n, h*m - h*h + 1 - delta[r, k, d]]]]; dmin) Table[uhat[n, 5, 2], {n, 1, 70}]
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