A105594 -id:A105594 - cp's OEIS frontend

A105595 Row sums of number triangle A105594.

1, 1, 3, 1, 3, 3, 3, 1, 3, 3, 3, 3, 3, 7, 3, 9, 1, 3, 3, 5, 5, 7, 9, 3, 3, 7, 9, 15, 7, 13, 21, 13, 1, 1, 5, 1, 3, 5, 7, 11, 3, 3, 7, 13, 5, 11, 13, 19, 3, 7, 7, 7, 7, 19, 9, 25, 7, 13, 21, 23, 15, 27, 19, 39, 1, 1, 3, 1, 5, 5, 3, 9, 3, 5, 11, 9, 9, 13, 19, 21, 3, 7, 7, 9, 7, 15, 17, 19, 7, 15, 21

Offset: 0

Paul Barry, Apr 14 2005

Conjecture : all terms are odd.

Cf. A105596.

A105595 := proc(n)
    add(A105594(n,k),k=0..n) ;
end proc: # R. J. Mathar, Nov 28 2014

a(n)=sum{k=0..n, mod(sum{j=0..n, abs(mu(binomial(n, j)))*mod(binomial(j, k), 2)}, 2)}

A105600 Assume the conjectured terms of A105594 are the correct beginnings of the trajectories described in A003508. a(n) is a record length of b(n) iterations to arrive at the collected trajectories. This sequence cites the a(n)'s.

Original entry on oeis.org

1, 5, 9, 16, 25, 43, 91, 105, 427, 463, 484, 4085, 4306, 4413, 5583, 6273, 10172, 18105, 24946, 31686, 31886

Offset: 0

R. K. Guy and Robert G. Wilson v, Apr 15 2005

nonn

The trajectory in A003508, etc., is defined as a(1)=n, for n>1, a(n) = a(n-1) + 1 + sum of distinct prime factors of a(n-1) that are < a(n-1).

Cf. A105593, the b(n)'s are in A105600.

a[1] = 1; a[n_] := a[n] = a[n - 1] + 1 + Plus @@ Select[ Flatten[ Table[ #[[1]], {1}] & /@ FactorInteger[ a[n - 1]]], # < a[n - 1] &]; t = Table[ a[n], {n, 1500}]; f[n_] := Module[{b, k = 1}, b[1] = n; b[m_] := b[m] = b[m - 1] + 1 + Plus @@ Select[ Flatten[ Table[ #[[1]], {1}] & /@ FactorInteger[ b[m - 1]]], # < b[m - 1] &]; While[ Position[t, b[k]] == {} && k < 1000, k++ ]; If[ k == 1000, t = Select[ Union[ Join[t, Table[ b[i], {i, 2, k}]]], # > n &]; -1, k - 1]]; lst = {{1, 0}}; Do[d = f[n]; If[d > lst[[ -1, 2]], AppendTo[lst, {n, d}]], {n, 60000}]; Transpose[ lst][[1]]

A105601 Assume the conjectured terms of A105594 are the correct beginnings of the trajectories described in A003508. a(n) is a record length of b(n) iterations to arrive at the collected trajectories. This sequence cites the b(n)'s.

Original entry on oeis.org

0, 2, 3, 7, 8, 12, 23, 40, 53, 54, 56, 72, 82, 113, 124, 129, 213, 214, 215, 216, 220

Offset: 0

R. K. Guy and Robert G. Wilson v, Apr 15 2005

nonn

The trajectory in A003508, etc., is defined as a(1)=n, for n>1, a(n) = a(n-1) + 1 + sum of distinct prime factors of a(n-1) that are < a(n-1).

Cf. A105593, the a(n)'s are in A105600.

a[1] = 1; a[n_] := a[n] = a[n - 1] + 1 + Plus @@ Select[ Flatten[ Table[ #[[1]], {1}] & /@ FactorInteger[ a[n - 1]]], # < a[n - 1] &]; t = Table[ a[n], {n, 1500}]; f[n_] := Module[{b, k = 1}, b[1] = n; b[m_] := b[m] = b[m - 1] + 1 + Plus @@ Select[ Flatten[ Table[ #[[1]], {1}] & /@ FactorInteger[ b[m - 1]]], # < b[m - 1] &]; While[ Position[t, b[k]] == {} && k < 1000, k++ ]; If[ k == 1000, t = Select[ Union[ Join[t, Table[ b[i], {i, 2, k}]]], # > n &]; -1, k - 1]]; lst = {{1, 0}}; Do[d = f[n]; If[d > lst[[ -1, 2]], AppendTo[lst, {n, d}]], {n, 60000}]; Transpose[ lst][[2]]

A105596 a(n) = Sum_{k=0..n} A105595(k)(-1)^kA105595(n-k) (interpolated zeros suppressed).

Original entry on oeis.org

1, 5, 13, 17, 25, 25, 33, 21, 9, -15, -23, -3, -11, -31, -47, -35, 5, -47, -83, -75, -211, -295, -267, -267, -99, -107, -159, -415, -347, -679, -279, -583, -395, -839, -1031, -1291, -1139, -1883, -1519, -1643, -855, -1591, -1571, -1851, -1195, -2419, -1923, -2179, -891, -1919, -2535

Offset: 0

Paul Barry, Apr 14 2005

Conjecture : a(2n)=1 mod 4 for all n, a(2n+1)=0 for all n.

Cf. A105594, A105595.

A105596 := proc(n)
    add(A105595(k)*(-1)^k*A105595(2*n-k),k=0..2*n) ;
end proc:
seq(A105596(n),n=0..50) ; # R. J. Mathar, Nov 28 2014

A105594[n_, k_] := A105594[n, k] = Sum[Abs[MoebiusMu[ Binomial[n, j]]*Mod[Binomial[j, k], 2]], {j, 0, n}]//Mod[#, 2]&;
A105595[n_] := Sum[A105594[n, k], {k, 0, n}];
a[n_] := Sum[A105595[k]*(-1)^k*A105595[2n - k], {k, 0, 2n}];
Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Aug 01 2023, after R. J. Mathar *)

A105597 Central numbers in a Moebius-binomial triangle.

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 0

Paul Barry, Apr 14 2005

Central numbers in A105595.
There seems to be a typo in above comment, maybe A105594 was intended? - Antti Karttunen, Aug 27 2017
Partial sums are A105598.

Antti Karttunen, Table of n, a(n) for n = 0..65537
Index entries for characteristic functions

a[n_]:= Binomial[Abs[MoebiusMu[n]],Abs[MoebiusMu[Floor[n/2]]]];Table[a[n],{n,0,100}] (* James C. McMahon, Jan 23 2024 *)

A105597(n) = if(n<2,1,binomial(abs(moebius(n)), abs(moebius(n\2)))); \\ Antti Karttunen, Aug 27 2017

a(n) = binomial(abs(mu(n)), abs(mu(floor(n/2)))).

cp's OEIS Frontend

A105595 Row sums of number triangle A105594.

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A105600 Assume the conjectured terms of A105594 are the correct beginnings of the trajectories described in A003508. a(n) is a record length of b(n) iterations to arrive at the collected trajectories. This sequence cites the a(n)'s.

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A105601 Assume the conjectured terms of A105594 are the correct beginnings of the trajectories described in A003508. a(n) is a record length of b(n) iterations to arrive at the collected trajectories. This sequence cites the b(n)'s.

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Mathematica

A105596 a(n) = Sum_{k=0..n} A105595(k)*(-1)^k*A105595(n-k) (interpolated zeros suppressed).

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Programs

Maple

Mathematica

A105597 Central numbers in a Moebius-binomial triangle.

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PARI

Formula

A105596 a(n) = Sum_{k=0..n} A105595(k)(-1)^kA105595(n-k) (interpolated zeros suppressed).