A289441 -id:A289441 - cp's OEIS frontend

A289439 The arithmetic function v_1(n,5).

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

Offset: 2

N. J. A. Sloane, Jul 07 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. A289440, A289441.

a:= n-> n*max(seq((floor((d-2)/5)+1)/d, d=numtheory[divisors](n))):
seq(a(n), n=2..100);  # Alois P. Heinz, Jul 07 2017

a[n_]:=n*Max[Table[(Floor[(d - 2)/5] + 1)/d, {d, Divisors[n]}]]; Table[a[n], {n, 2, 100}] (* Indranil Ghosh, Jul 08 2017 *)

v(g,n,h)={my(t=0);fordiv(n,d,t=max(t,((d-1-gcd(d,g))\h + 1)*(n/d)));t}
a(n)=v(1,n,5); \\ Andrew Howroyd, Jul 07 2017

from sympy import divisors, floor
def a(n): return int(n*max(int(floor((d - 2)/5) + 1)/d for d in divisors(n)))
print([a(n) for n in range(2, 101)]) # Indranil Ghosh, Jul 08 2017

a(41)-a(70) from Andrew Howroyd, Jul 07 2017

A289440 The arithmetic function v_3(n,5).

Original entry on oeis.org

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

Offset: 2

N. J. A. Sloane, Jul 07 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. A289439, A289441.

a:= n-> n*max(seq((floor((d-1-igcd(d, 3))/5)+1)
        /d, d=numtheory[divisors](n))):
seq(a(n), n=2..100);  # Alois P. Heinz, Jul 07 2017

a[n_]:=n*Max[Table[(Floor[(d - 1 - GCD[d, 3])/5] + 1)/d, {d, Divisors[n]}]]; Table[a[n], {n, 2, 100}] (* Indranil Ghosh, Jul 08 2017 *)

v(g,n,h)={my(t=0);fordiv(n,d,t=max(t,((d-1-gcd(d,g))\h + 1)*(n/d)));t}
a(n)=v(3,n,5); \\ Andrew Howroyd, Jul 07 2017

from sympy import divisors, floor, gcd
def a(n): return n*max([(floor((d - 1 - gcd(d, 3))/5) + 1)/d for d in divisors(n)])
print([a(n) for n in range(2, 101)]) # Indranil Ghosh, Jul 08 2017

a(41)-a(70) from Andrew Howroyd, Jul 07 2017

A290988 The arithmetic function v+-(n,3).

Original entry on oeis.org

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

Offset: 2

Robert Price, Aug 16 2017

nonn

Bela Bajnok, Additive Combinatorics: A Menu of Research Problems, arXiv:1705.07444 [math.NT], May 2017. See Section 1.6.2.

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

vpm[n_, h_] := (d = Divisors[n]; Max[(2*Floor[(d - 2)/(2*h)] + 1)*n/d]); Table[vpm[n, 3], {n, 2, 70}]

A289187 The arithmetic function v_1(n,6).

Original entry on oeis.org

1, 1, 2, 1, 3, 1, 4, 3, 5, 2, 6, 2, 7, 5, 8, 3, 9, 3, 10, 7, 11, 4, 12, 5, 13, 9, 14, 5, 15, 5, 16, 11, 17, 7, 18, 6, 19, 13, 20, 7, 21, 7, 22, 15, 23, 8, 24, 8, 25, 17, 26, 9, 27, 11, 28, 19, 29, 10, 30, 10, 31, 21, 32, 13, 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[1, n, 6], {n, 2, 70}]
a[n_]:=n Max[Table[(Floor[(d - 1 - GCD[d, 1])/6] + 1)/d, {d, Divisors[n]}]]; Table[a[n], {n, 2, 100}] (* Vincenzo Librandi, Aug 17 2017 *)

A291041 The arithmetic function uhat(n,7,4).

Original entry on oeis.org

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

Offset: 1

Robert Price, Aug 16 2017

nonn

The sequence appears to be equal to uhat(n,7,3), at least to 10000 terms.

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, 7, 4], {n, 1, 70}]

A290636 The arithmetic function v_3(n,4).

Original entry on oeis.org

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

Offset: 2

Robert Price, Aug 13 2017

nonn

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

Robert Israel, Table of n, a(n) for n = 2..10000
Bela Bajnok, Additive Combinatorics: A Menu of Research Problems, arXiv:1705.07444 [math.NT], May 2017. See Table in Section 1.6.1.

Cf. A211316 (equals v_1(n,3)).
Cf. A289435 - A289441.
Cf. A024698.

f:= n -> max(map(t -> (floor((t - 1 - igcd(t, 3))/4) + 1)*n/t, numtheory:-divisors(n))):
map(f, [$2..100]); # Robert Israel, Aug 16 2017

v[g_, n_, h_] := (d = Divisors[n]; Max[(Floor[(d - 1 - GCD[d, g])/h] + 1)*n/d]); Table[v[3, n, 4], {n, 2, 70}]

For n >= 3, f(prime(n)) = A024698(n-1). - Robert Israel, Aug 16 2017

A291306 The arithmetic function v_6(n,1).

Original entry on oeis.org

0, 0, 2, 4, 0, 6, 6, 6, 8, 10, 6, 12, 12, 12, 14, 16, 12, 18, 18, 18, 20, 22, 18, 24, 24, 24, 26, 28, 24, 30, 30, 30, 32, 34, 30, 36, 36, 36, 38, 40, 36, 42, 42, 42, 44, 46, 42, 48, 48, 48, 50, 52, 48, 54, 54, 54, 56, 58, 54, 60, 60, 60, 62, 64, 60, 66, 66, 66, 68

Offset: 2

Robert Price, Aug 21 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.

Cf. A089128.

seq(n-gcd(n,6), n=2..100); # Ridouane Oudra, Dec 15 2024

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

a(n) = n - gcd(n,6) = n - A089128(n). - Ridouane Oudra, Dec 15 2024

Sum_{n>=7} (-1)^n/a(n) = Pi/(6*sqrt(3)) - 1/4. - Amiram Eldar, Jan 15 2025

A291330 The arithmetic function v_4(n,1).

Original entry on oeis.org

0, 2, 0, 4, 4, 6, 4, 8, 8, 10, 8, 12, 12, 14, 12, 16, 16, 18, 16, 20, 20, 22, 20, 24, 24, 26, 24, 28, 28, 30, 28, 32, 32, 34, 32, 36, 36, 38, 36, 40, 40, 42, 40, 44, 44, 46, 44, 48, 48, 50, 48, 52, 52, 54, 52, 56, 56, 58, 56, 60, 60, 62, 60, 64, 64, 66, 64, 68, 68

Offset: 2

Robert Price, Aug 22 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.

Cf. A109008.

seq(n-gcd(n,4), n=2..100); # Ridouane Oudra, Dec 15 2024

v[g_, n_, h_] := (d = Divisors[n]; Max[(Floor[(d - 1 - GCD[d, g])/h] + 1)*n/d]); Table[v[4, n, 1], {n, 2, 70}]

/* Adapted from Mathematica program */
v(g, n, h) = my(d=divisors(n)); for(k=1, #d, d[k]=floor(((d[k]-1-gcd(d[k], g))/h) + 1)*n/d[k]); vecmax(d)
a(n) = v(4, n, 1) \\ Felix Fröhlich, Aug 22 2017

a(n) = n - gcd(n,4) = n - A109008(n). - Ridouane Oudra, Dec 15 2024

Sum_{n>=5} (-1)^n/a(n) = (1 - log(2))/2. - Amiram Eldar, Jan 15 2025

A291357 The arithmetic function u(n,2,3).

Original entry on oeis.org

4, 2, 3, 2, 4, 2, 4, 2, 3, 2, 4, 2, 4, 2, 3, 2, 4, 2, 4, 2, 3, 2, 4, 2, 4, 2, 3, 2, 4, 2, 4, 2, 3, 2, 4, 2, 4, 2, 3, 2, 4, 2, 4, 2, 3, 2, 4, 2, 4, 2, 3, 2, 4, 2, 4, 2, 3, 2, 4, 2, 4, 2, 3, 2, 4, 2, 4, 2, 3, 2, 4, 2, 4, 2, 3, 2, 4, 2, 4, 2, 3, 2, 4, 2, 4, 2, 3, 2, 4, 2, 4, 2, 3, 2, 4, 2, 4, 2, 3, 2, 4, 2, 4, 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, 3], {n, 1, 70}]

A291357(n, m=2, h=3) = { 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

A291358 The arithmetic function u(n,2,4).

Original entry on oeis.org

5, 2, 3, 2, 5, 2, 5, 2, 3, 2, 5, 2, 5, 2, 3, 2, 5, 2, 5, 2, 3, 2, 5, 2, 5, 2, 3, 2, 5, 2, 5, 2, 3, 2, 5, 2, 5, 2, 3, 2, 5, 2, 5, 2, 3, 2, 5, 2, 5, 2, 3, 2, 5, 2, 5, 2, 3, 2, 5, 2, 5, 2, 3, 2, 5, 2, 5, 2, 3, 2, 5, 2, 5, 2, 3, 2, 5, 2, 5, 2, 3, 2, 5, 2, 5, 2, 3, 2, 5, 2, 5, 2, 3, 2, 5, 2, 5, 2, 3, 2, 5, 2, 5, 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, 4], {n, 1, 70}]

A291358(n, m=2, h=4) = { 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

cp's OEIS Frontend

A289439 The arithmetic function v_1(n,5).

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A289440 The arithmetic function v_3(n,5).

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A290988 The arithmetic function v+-(n,3).

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A289187 The arithmetic function v_1(n,6).

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A291041 The arithmetic function uhat(n,7,4).

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A290636 The arithmetic function v_3(n,4).

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A291306 The arithmetic function v_6(n,1).

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A291330 The arithmetic function v_4(n,1).

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