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.

Previous Showing 101-110 of 113 results. Next

A291568 The arithmetic function uhat(n,5,5).

Original entry on oeis.org

1, 1, 1, 1, -3, 1, 1, 1, 1, -3, 1, 1, 1, 1, -3, 1, 1, 1, 1, -3, 1, 1, 1, 1, -3, 1, 1, 1, 1, -3, 1, 1, 1, 1, -3, 1, 1, 1, 1, -3, 1, 1, 1, 1, -3, 1, 1, 1, 1, -3, 1, 1, 1, 1, -3, 1, 1, 1, 1, -3, 1, 1, 1, 1, -3, 1, 1, 1, 1, -3
Offset: 1

Views

Author

Robert Price, Aug 26 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, 5,5], {n, 1, 70}]

Formula

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

A291569 The arithmetic function uhat(n,5,7).

Original entry on oeis.org

-13, -13, -13, -13, -15, -13, -13, -13, -13, -15, -13, -13, -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 26 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, 5, 7], {n, 1, 70}]

A291570 The arithmetic function uhat(n,5,8).

Original entry on oeis.org

-23, -23, -23, -23, -25, -24, -23, -23, -23, -25, -23, -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 26 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, 5, 8], {n, 1, 70}]

A291571 The arithmetic function uhat(n,6,2).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 8, 9, 6, 9, 7, 9, 8, 9, 6, 9, 8, 7, 8, 9, 6, 9, 8, 9, 7, 9, 6, 9, 8, 9, 8, 7, 6, 9, 8, 9, 8, 9, 6, 9, 8, 9, 8, 9, 6, 7, 8, 9, 8, 9, 6, 9, 7, 9, 8, 9, 6, 9, 8, 7, 8, 9, 6, 9, 8, 9, 7
Offset: 1

Views

Author

Robert Price, Aug 26 2017

Keywords

Comments

uhat(6,4) also generates this sequence.

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, 6,2], {n, 1, 70}]

A291572 The arithmetic function uhat(n,6,3).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 8, 10, 10, 6, 10, 7, 8, 8, 10, 6, 10, 10, 7, 10, 10, 6, 10, 10, 8, 7, 10, 6, 10, 8, 8, 10, 7, 6, 10, 10, 8, 8, 10, 6, 10, 10, 8, 10, 10, 6, 7, 10, 8, 10, 10, 6, 10, 7, 8, 10, 10, 6, 10, 10, 7, 8, 10, 6, 10, 10, 8, 7
Offset: 1

Views

Author

Robert Price, Aug 26 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, 6, 3], {n, 1, 70}]

A291574 The arithmetic function uhat(n,6,6).

Original entry on oeis.org

1, 0, -1, 0, 1, -4, 1, 0, -1, 0, 1, -4, 1, 0, -1, 0, 1, -4, 1, 0, -1, 0, 1, -4, 1, 0, -1, 0, 1, -4, 1, 0, -1, 0, 1, -4, 1, 0, -1, 0, 1, -4, 1, 0, -1, 0, 1, -4, 1, 0, -1, 0, 1, -4, 1, 0, -1, 0, 1, -4, 1, 0, -1, 0, 1, -4, 1, 0, -1, 0
Offset: 1

Views

Author

Robert Price, Aug 26 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, 6, 6], {n, 1, 70}]

Formula

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

A291575 The arithmetic function uhat(n,6,8).

Original entry on oeis.org

-15, -16, -15, -16, -15, -18, -15, -16, -15, -16, -15, -18, -15, -16, -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 26 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, 6, 8], {n, 1, 70}]

A291576 The arithmetic function uhat(n,7,2).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 11, 11, 7, 11, 8, 11, 9, 11, 10, 7, 11, 11, 8, 11, 11, 9, 7, 11, 10, 11, 8, 11, 11, 7, 9, 11, 11, 11, 8, 11, 7, 11, 11, 9, 11, 11, 8, 7, 10, 11, 11, 11, 9, 11, 7, 11, 11, 11, 10, 11, 11, 7, 8, 11, 11, 11, 11, 11, 7
Offset: 1

Views

Author

Robert Price, Aug 26 2017

Keywords

Comments

uhat(7,5) also generates this sequence.

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, 7, 2], {n, 1, 70}]

A291578 The arithmetic function uhat(n,7,7).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, -5, 1, 1, 1, 1, 1, 1, -5, 1, 1, 1, 1, 1, 1, -5, 1, 1, 1, 1, 1, 1, -5, 1, 1, 1, 1, 1, 1, -5, 1, 1, 1, 1, 1, 1, -5, 1, 1, 1, 1, 1, 1, -5, 1, 1, 1, 1, 1, 1, -5, 1, 1, 1, 1, 1, 1, -5, 1, 1, 1, 1, 1, 1, -5
Offset: 1

Views

Author

Robert Price, Aug 26 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, 7, 7], {n, 1, 70}]

A291579 The arithmetic function uhat(n,8,2).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 12, 13, 8, 13, 9, 13, 10, 13, 11, 13, 8, 13, 12, 9, 12, 13, 10, 13, 8, 11, 12, 13, 9, 13, 12, 13, 8, 13, 12, 13, 11, 9, 12, 13, 8, 13, 10, 13, 12, 13, 9, 11, 8, 13, 12, 13, 10, 13, 12, 9, 8, 13, 11, 13, 12, 13, 10
Offset: 1

Views

Author

Robert Price, Aug 26 2017

Keywords

Comments

uhat(8,6) also generates this sequence.

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, 8, 2], {n, 1, 70}]
Previous Showing 101-110 of 113 results. Next

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