A289438 -id:A289438 - cp's OEIS frontend

A291361 The arithmetic function u(n,2,6).

7, 2, 3, 2, 5, 2, 7, 2, 3, 2, 7, 2, 7, 2, 3, 2, 7, 2, 7, 2, 3, 2, 7, 2, 5, 2, 3, 2, 7, 2, 7, 2, 3, 2, 5, 2, 7, 2, 3, 2, 7, 2, 7, 2, 3, 2, 7, 2, 7, 2, 3, 2, 7, 2, 5, 2, 3, 2, 7, 2, 7, 2, 3, 2, 5, 2, 7, 2, 3, 2, 7, 2, 7, 2, 3, 2, 7, 2, 7, 2, 3, 2, 7, 2, 5, 2, 3, 2, 7, 2, 7, 2, 3, 2, 5, 2, 7, 2, 3, 2, 7, 2, 7, 2, 3

Offset: 1

Robert Price, Aug 23 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537
Bela Bajnok, Additive Combinatorics: A Menu of Research Problems, arXiv:1705.07444 [math.NT], May 2017. See Table in Section 1.6.3.

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

u[n_, m_, h_] := (d = Divisors[n]; Min[(h*Ceiling[m/d] - h + 1)*d]); Table[u[n, 2, 6], {n, 1, 70}]

A291361(n, m=2, h=6) = { my(p=0,k); fordiv(n,d,k = d*(h*ceil(m/d) - h + 1); if(!p || (k < p), p = k)); (p); }; \\ Antti Karttunen, Oct 01 2018

More terms from Antti Karttunen, Oct 01 2018

A291362 The arithmetic function u(n,2,7).

Original entry on oeis.org

8, 2, 3, 2, 5, 2, 7, 2, 3, 2, 8, 2, 8, 2, 3, 2, 8, 2, 8, 2, 3, 2, 8, 2, 5, 2, 3, 2, 8, 2, 8, 2, 3, 2, 5, 2, 8, 2, 3, 2, 8, 2, 8, 2, 3, 2, 8, 2, 7, 2, 3, 2, 8, 2, 5, 2, 3, 2, 8, 2, 8, 2, 3, 2, 5, 2, 8, 2, 3, 2, 8, 2, 8, 2, 3, 2, 7, 2, 8, 2, 3, 2, 8, 2, 5, 2, 3, 2, 8, 2, 7, 2, 3, 2, 5, 2, 8, 2, 3, 2, 8, 2, 8, 2, 3

Offset: 1

Robert Price, Aug 23 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537
Bela Bajnok, Additive Combinatorics: A Menu of Research Problems, arXiv:1705.07444 [math.NT], May 2017. See Table in Section 1.6.3.

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

u[n_, m_, h_] := (d = Divisors[n]; Min[(h*Ceiling[m/d] - h + 1)*d]); Table[u[n, 2, 7], {n, 1, 70}]

A291362(n, m=2, h=7) = { my(p=0,k); fordiv(n,d,k = d*(h*ceil(m/d) - h + 1); if(!p || (k < p), p = k)); (p); }; \\ Antti Karttunen, Oct 01 2018

More terms from Antti Karttunen, Oct 01 2018

A291363 The arithmetic function u(n,2,8).

Original entry on oeis.org

9, 2, 3, 2, 5, 2, 7, 2, 3, 2, 9, 2, 9, 2, 3, 2, 9, 2, 9, 2, 3, 2, 9, 2, 5, 2, 3, 2, 9, 2, 9, 2, 3, 2, 5, 2, 9, 2, 3, 2, 9, 2, 9, 2, 3, 2, 9, 2, 7, 2, 3, 2, 9, 2, 5, 2, 3, 2, 9, 2, 9, 2, 3, 2, 5, 2, 9, 2, 3, 2, 9, 2, 9, 2, 3, 2, 7, 2, 9, 2, 3, 2, 9, 2, 5, 2, 3, 2, 9, 2, 7, 2, 3, 2, 5, 2, 9, 2, 3, 2, 9, 2, 9, 2, 3

Offset: 1

Robert Price, Aug 23 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537
Bela Bajnok, Additive Combinatorics: A Menu of Research Problems, arXiv:1705.07444 [math.NT], May 2017. See Table in Section 1.6.3.

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

u[n_, m_, h_] := (d = Divisors[n]; Min[(h*Ceiling[m/d] - h + 1)*d]); Table[u[n, 2, 8], {n, 1, 70}]

A291363(n, m=2, h=8) = { my(p=0,k); fordiv(n,d,k = d*(h*ceil(m/d) - h + 1); if(!p || (k < p), p = k)); (p); }; \\ Antti Karttunen, Oct 01 2018

More terms from Antti Karttunen, Oct 01 2018

A291514 The arithmetic function uhat(n,3,5).

Original entry on oeis.org

-9, -9, -9, -9, -9, -9, -9, -9, -9, -10, -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

Robert Price, Aug 25 2017

sign

Bela Bajnok, Additive Combinatorics: A Menu of Research Problems, arXiv:1705.07444 [math.NT], May 2017. See Table in Section 1.6.4.

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

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, 5], {n, 1, 70}]

Conjectures from Chai Wah Wu, Jun 10 2025: (Start)
a(n) = 2*a(n-1) - a(n-2) for n > 10.
G.f.: x*(-x^9 + 9*x - 9)/(x - 1)^2. (End)

A291520 The arithmetic function uhat(n,4,2).

Original entry on oeis.org

1, 2, 3, 4, 5, 4, 5, 4, 5, 4, 5, 4, 5, 4, 5, 4, 5, 4, 5, 4, 5, 4, 5, 4, 5, 4, 5, 4, 5, 4, 5, 4, 5, 4, 5, 4, 5, 4, 5, 4, 5, 4, 5, 4, 5, 4, 5, 4, 5, 4, 5, 4, 5, 4, 5, 4, 5, 4, 5, 4, 5, 4, 5, 4, 5, 4, 5, 4, 5, 4

Offset: 1

Robert Price, Aug 25 2017

nonn

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Bela Bajnok, Additive Combinatorics: A Menu of Research Problems, arXiv:1705.07444 [math.NT], May 2017. See Table in Section 1.6.4.

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

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

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

A289186 The arithmetic function v_4(n,5).

Original entry on oeis.org

0, 1, 0, 1, 2, 2, 1, 3, 2, 2, 4, 3, 4, 5, 3, 4, 6, 4, 4, 7, 4, 5, 8, 5, 6, 9, 8, 6, 10, 6, 6, 11, 8, 10, 12, 8, 8, 13, 8, 8, 14, 9, 8, 15, 10, 10, 16, 14, 10, 17, 12, 11, 18, 11, 16, 19, 12, 12, 20, 12, 12, 21, 12, 15, 22, 14, 16, 23, 20

Offset: 2

Robert Price, Aug 16 2017

nonn

J. Butterworth, Examining the arithmetic function v_g(n,h). Research Papers in Mathematics, B. Bajnok, ed., Gettysburg College, Vol. 8 (2008).

Bela Bajnok, Additive Combinatorics: A Menu of Research Problems, arXiv:1705.07444 [math.NT], May 2017. See Table in Section 1.6.1.

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

v[g_, n_, h_] := (d = Divisors[n]; Max[(Floor[(d - 1 - GCD[d, g])/h] + 1)*n/d]); Table[v[4, n, 5], {n, 2, 70}]
a[n_]:=n*Max[Table[(Floor[(d - 1 - GCD[d, 4])/5] + 1)/d, {d, Divisors[n]}]]; Table[a[n], {n, 2, 100}] (* Vincenzo Librandi, Aug 17 2017 *)

A290978 The arithmetic function v_2(n,5).

Original entry on oeis.org

0, 1, 1, 1, 2, 2, 2, 3, 2, 2, 4, 3, 4, 5, 4, 4, 6, 4, 5, 7, 4, 5, 8, 5, 6, 9, 8, 6, 10, 6, 8, 11, 8, 10, 12, 8, 8, 13, 10, 8, 14, 9, 11, 15, 10, 10, 16, 14, 10, 17, 13, 11, 18, 11, 16, 19, 12, 12, 20, 12, 12, 21, 16, 15, 22, 14, 17, 23, 20

Offset: 2

Robert Price, Aug 16 2017

nonn

J. Butterworth, Examining the arithmetic function v_g(n,h). Research Papers in Mathematics, B. Bajnok, ed., Gettysburg College, Vol. 8 (2008).

Bela Bajnok, Additive Combinatorics: A Menu of Research Problems, arXiv:1705.07444 [math.NT], May 2017. See Table in Section 1.6.1.

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

v[g_, n_, h_] := (d = Divisors[n]; Max[(Floor[(d - 1 - GCD[d, g])/h] + 1)*n/d]); Table[v[2, n, 5], {n, 2, 70}]
a[n_]:=n Max[Table[(Floor[(d - 1 - GCD[d, 2])/5] + 1)/d, {d, Divisors[n]}]]; Table[a[n], {n, 2, 100}] (* Vincenzo Librandi, Aug 17 2017 *)

A290979 The arithmetic function v_2(n,6).

Original entry on oeis.org

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

Offset: 2

Robert Price, Aug 16 2017

nonn

J. Butterworth, Examining the arithmetic function v_g(n,h). Research Papers in Mathematics, B. Bajnok, ed., Gettysburg College, Vol. 8 (2008).

Bela Bajnok, Additive Combinatorics: A Menu of Research Problems, arXiv:1705.07444 [math.NT], May 2017. See Table in Section 1.6.1.

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

v[g_, n_, h_] := (d = Divisors[n]; Max[(Floor[(d - 1 - GCD[d, g])/h] + 1)*n/d]); Table[v[2, n, 6], {n, 2, 70}]
a[n_]:=n Max[Table[(Floor[(d - 1 - GCD[d, 2])/6] + 1)/d, {d, Divisors[n]}]]; Table[a[n], {n, 2, 100}] (* Vincenzo Librandi, Aug 19 2017 *)

A290981 The arithmetic function v_4(n,6).

Original entry on oeis.org

0, 1, 0, 1, 2, 1, 1, 3, 2, 2, 4, 2, 2, 5, 2, 3, 6, 3, 4, 7, 4, 4, 8, 5, 4, 9, 4, 5, 10, 5, 5, 11, 6, 7, 12, 6, 6, 13, 8, 7, 14, 7, 8, 15, 8, 8, 16, 8, 10, 17, 8, 9, 18, 11, 9, 19, 10, 10, 20, 10, 10, 21, 10, 13, 22, 11, 12, 23, 14

Offset: 2

Robert Price, Aug 16 2017

nonn

J. Butterworth, Examining the arithmetic function v_g(n,h). Research Papers in Mathematics, B. Bajnok, ed., Gettysburg College, Vol. 8 (2008).

Bela Bajnok, Additive Combinatorics: A Menu of Research Problems, arXiv:1705.07444 [math.NT], May 2017. See Table in Section 1.6.1.

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

v[g_, n_, h_] := (d = Divisors[n]; Max[(Floor[(d - 1 - GCD[d, g])/h] + 1)*n/d]); Table[v[4, n, 6], {n, 2, 70}]
a[n_]:=n Max[Table[(Floor[(d - 1 - GCD[d, 4])/6] + 1)/d, {d, Divisors[n]}]]; Table[a[n], {n, 2, 100}] (* Vincenzo Librandi, Aug 19 2017 *)

A290982 The arithmetic function v_5(n,6).

Original entry on oeis.org

1, 1, 2, 0, 3, 1, 4, 3, 5, 2, 6, 2, 7, 5, 8, 3, 9, 3, 10, 7, 11, 4, 12, 4, 13, 9, 14, 5, 15, 5, 16, 11, 17, 5, 18, 6, 19, 13, 20, 7, 21, 7, 22, 15, 23, 8, 24, 8, 25, 17, 26, 9, 27, 10, 28, 19, 29, 10, 30, 10, 31, 21, 32, 10, 33, 11, 34, 23, 35

Offset: 2

Robert Price, Aug 16 2017

nonn

J. Butterworth, Examining the arithmetic function v_g(n,h). Research Papers in Mathematics, B. Bajnok, ed., Gettysburg College, Vol. 8 (2008).

Bela Bajnok, Additive Combinatorics: A Menu of Research Problems, arXiv:1705.07444 [math.NT], May 2017. See Table in Section 1.6.1.

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

v[g_, n_, h_] := (d = Divisors[n]; Max[(Floor[(d - 1 - GCD[d, g])/h] + 1)*n/d]); Table[v[5, n, 6], {n, 2, 70}]
a[n_]:=n Max[Table[(Floor[(d - 1 - GCD[d, 5])/6] + 1)/d, {d, Divisors[n]}]]; Table[a[n], {n, 2, 100}] (* Vincenzo Librandi, Aug 19 2017 *)

cp's OEIS Frontend

A291361 The arithmetic function u(n,2,6).

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A291362 The arithmetic function u(n,2,7).

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A291363 The arithmetic function u(n,2,8).

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A291514 The arithmetic function uhat(n,3,5).

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A291520 The arithmetic function uhat(n,4,2).

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A289186 The arithmetic function v_4(n,5).

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A290978 The arithmetic function v_2(n,5).

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A290979 The arithmetic function v_2(n,6).

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A290981 The arithmetic function v_4(n,6).

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