0,7

Table of n, a(n) for n=0..44.

Conjecture: a(n) ~ sqrt(3) * 2^(n+1) / (sqrt(Pi) * n^(7/2)). - Vaclav Kotesovec, Dec 29 2022

Table[Max[CoefficientList[Product[1+x^(k^3), {k, n}], x]], {n, 0, 44}] (* Stefano Spezia, Dec 25 2022 *)

nmax = 100; poly = ConstantArray[0, nmax^2*(nmax + 1)^2/4 + 1]; poly[[1]] = 1; poly[[2]] = 1; Do[Do[poly[[j + 1]] += poly[[j - k^3 + 1]], {j, k^2*(k + 1)^2/4, k^3, -1}]; Print[k, " ", Max[poly]], {k, 2, nmax}]; (* Vaclav Kotesovec, Dec 29 2022 *)

(PARI) a(n) = vecmax(Vec(prod(i=1, n, (1+x^(i^3))))); \\ Michel Marcus, Dec 27 2022

Cf. A000537, A000578, A025591, A160235, A279329, A359320.

nonn,new

Ilya Gutkovskiy, Dec 25 2022

approved

0,10

Table of n, a(n) for n=0..50.

f:= proc(n) local i; max(coeffs(expand(mul(1+x^(i^4), i=1..n)))) end proc:

map(f, [$1..50]); # Robert Israel, Dec 26 2022

(PARI) a(n) = vecmax(Vec(prod(k=1, n, 1+x^(k^4)))); \\ Michel Marcus, Dec 26 2022

Cf. A000538, A000583, A025591, A160235, A298859, A359319.

nonn,new

Ilya Gutkovskiy, Dec 25 2022

a(38)-a(50) from Seiichi Manyama, Dec 26 2022

approved

1,2

For n>1, the order of the twisted Chevalley group (3)D_4(q).

R. W. Carter, Simple Groups of Lie Type, Wiley 1972, Chap. 14, p. 262.

J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, ATLAS of Finite Groups. Oxford Univ. Press, 1985, p. xvi.

Table of n, a(n) for n=1..10.

Cf. A000961, A037253.

nonn,new

N. J. A. Sloane, Dec 28 2022

approved

1,2

There is a second version of this sequence possible if we change the definition to a(1) = 2 and a(n) > 1, then the sequence will start 2, 4, 5, 8, 10, 7, 14, ... . It will after this continue in the same way as our actual sequence does (and would also extend the valid range of the recurrence formulas).

Start a(1) = 2 and value 1 allowed is A257794.

Table of n, a(n) for n=1..64.

Index entries for linear recurrences with constant coefficients, signature (0,0,1,0,0,1,0,0,-1).

G.f.: x*(1 + 3*x + 7*x^2 + 4*x^3 + 7*x^4 + x^5 + 8*x^6 + 3*x^7 - 4*x^8 + 2*x^9 - x^10 + x^11 - 3*x^12 + x^14)/(1 - x^3 - x^6 + x^9).

a(n) = a(n-3) + a(n-6) - a(n-9) for n >= 16.

a((3*(2*n-1) - (-1)^n)/4) = (3*(2*n-1) - (-1)^n)/2, for n > 3.

a(6*n) = 6*n+1, for n > 1.

a(6*n+3) = 6*n+5, for n > 0.

a(n) = 30*n - 2*a(n-1) - 3*a(n-2) - 3*a(n-3) - 3*a(n-4) - 3*a(n-5) - 2*a(n-6) - a(n-7) - 96, for n > 13.

(PARI) a(n) = {my(v = [1, 3, 7, 5, 10, 8]); if(n < 7, v[n], n*(1+min(1, n%3))+(n%3 == 0)+(n%6 == 3))}

Cf. A002516, A105753, A257794.

nonn,easy,new

Thomas Scheuerle, Dec 01 2022

approved

1

Table of n, a(n) for n=1..125.

a(n) = A166486(n) - A359370(n).

a(n) = [A358839(n) < 0], where [ ] is the Iverson bracket.

(PARI) A359372(n) = ((n%4)&&(bigomega(n)%2));

Characteristic function of A359373.

Cf. A001222, A166486, A358839, A359372.

nonn,new

Antti Karttunen, Dec 28 2022

approved

1,2

Seiichi Manyama, Table of n, a(n) for n = 1..1000

S:= 1 + x*(x^2 + 4*x + 1)/(x - 1)^4:

for n from 1 to 30 do

SS:= series(S, x, n+1);

A[n]:= coeff(SS, x, n);

S:= S/(1+A[n]*x^n);

od:

seq(A[i], i=1..30); # Robert Israel, Dec 28 2022

Cf. A000578, A147559, A147654, A316083.

sign,new

Seiichi Manyama, Dec 28 2022

approved

1

Note the correspondences between four sequences:

A355689 --- abs ---> A353627

^ ^

| |

D.I. D.I.

| |

v v

A166486 <--- abs --- A358839 (this sequence)

Here D.I. means that the sequences are Dirichlet Inverses of each other, and abs means taking absolute values.

Table of n, a(n) for n=1..105.

Multiplicative with a(p^e) = (-1)^e for odd primes p, and a(2^e) = -1 if e = 1, otherwise 0.

For all e >= 0, a(2^e) = A008683(2^e).

For all n >= 0, a(2n+1) = A008836(2n+1).

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, d<n} A353627(n/d) * a(d).

f[p_, e_] := (-1)^e; f[2, e_] := If[e == 1, -1, 0]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Dec 28 2022 *)

(PARI) A358839(n) = { my(f = factor(n)); prod(k=1, #f~, if(2==f[k, 1], -(1==f[k, 2]), (-1)^f[k, 2])); };

Cf. A008683, A008836.

Cf. A166486 (absolute values), A353627 (Dirichlet inverse), A355689 (Dirichlet inverse of the absolute values).

Cf. A008586 (after its initial term gives the positions of 0's), A359371 (positive terms), A359373 (negative terms).

sign,mult,new

Antti Karttunen, Dec 23 2022

approved

1

Table of n, a(n) for n=1..121.

Index entries for characteristic functions

a(n) = A166486(n) - A359372(n).

a(n) = [A358839(n) > 0], where [ ] is the Iverson bracket.

(PARI) A359370(n) = ((n%4)&&!(bigomega(n)%2));

Characteristic function of A359371.

Cf. A001222, A166486, A358839, A359372.

nonn,new

Antti Karttunen, Dec 28 2022

approved

1,2

1, and semiprimes other than 4, multiplied by a product of 0 or more odd semiprimes. - Robert Israel, Dec 28 2022

Table of n, a(n) for n=1..68.

{k | A008836(k) > 0 and A010873(k) > 0}.

select(t -> numtheory:-bigomega(t)::even, [seq(seq(4*i+j, j=1..3), i=0..100)]); # Robert Israel, Dec 28 2022

Select[Range[200], And[LiouvilleLambda[#] > 0, ! Divisible[#, 4]] &] (* Michael De Vlieger, Dec 28 2022 *)

(PARI) isA359371(n) = A359370(n);

Intersection of A028260 and A042968.

Setwise difference A042968 \ A359373.

Positions of positive terms in A358839.

Cf. A001222, A008836, A010873, A046337 (subsequence), A166486, A359370 (characteristic function).

nonn,new

Antti Karttunen, Dec 28 2022

approved

1,1

Table of n, a(n) for n=1..69.

{k | A008836(k) < 0 and A010873(k) > 0}.

Select[Range[200], And[LiouvilleLambda[#] < 0, ! Divisible[#, 4]] &] (* Michael De Vlieger, Dec 28 2022 *)

(PARI) isA359373(n) = A359372(n);

Intersection of A026424 and A042968.

Setwise difference A042968 \ A359371.

Positions of negative terms in A358839.

Cf. A001222, A008836, A010873, A067019 (subsequence), A166486, A359372 (characteristic function).

nonn,new

Antti Karttunen, Dec 28 2022

approved