A077597 -id:A077597 - cp's OEIS frontend

A077596 Central coefficients of Moebius polynomials (A074586): coefficient of x^(n/2-1/2) if n is odd; coefficient of x^(n/2-1) if n is even and >4. The n-th Moebius polynomial, M(n,x), satisfies M(n,-1)=mu(n) the Moebius function of n.

1, 2, 4, 8, 15, 30, 57, 108, 206, 393, 752, 1439, 2772, 5334, 10327, 19967, 38808, 75319, 146844, 285862, 558723, 1090370, 2135551, 4176224, 8193490, 16050930, 31537017, 61872863, 121721157, 239115024, 470918888, 926141652, 1825708221

Offset: 1

Benoit Cloitre and Paul D. Hanna, Nov 10 2002

nonn

These terms seem to be asymptotic to c*2^n/sqrt(n) with c=1.2208..

c = 1.220916104316909855089768170983761594215082355524... . - Vaclav Kotesovec, Feb 11 2015

			These are the largest coefficients of the Moebius polynomials, which begin:
M(1,x) = 1;
M(2,x) = 1 + 2x;
M(3,x) = 1 + 4x + 2x^2;
M(4,x) = 1 + 7x + 8x^2 + 2x^3;
M(5,x) = 1 + 9x +15x^2 +10x^3 + 2x^4;
M(6,x) = 1 +13x +30x^2 +27x^3 +12x^4 + 2x^5;
M(7,x) = 1 +15x +43x^2 +57x^3 +39x^4 +14x^5 + 2x^6;
M(8,x) = 1 +19x +67x^2+108x^3 +98x^4 +53x^5 +16x^6 + 2x^7; ...

Cf. A074586, A074587, A077597, A077598, A077599, A077600, A077601.

m[n_, 1] = 1; m[n_, k_] := m[n, k] = Sum[Floor[n/j]*m[j, k - 1], {j, 1, n - 1}];
a[n_ /; n <= 4] := 2^(n - 1); a[n_?OddQ] := m[n, (n + 1)/2]; a[n_?EvenQ] := m[n, n/2]; Table[a[n], {n, 1, 33}] (* Jean-François Alcover, Jun 18 2013 *)

A077598 Coefficient of x^2 in the n-th Moebius polynomial (A074586), M(n,x), which satisfies M(n,-1)=mu(n) the Moebius function of n.

Original entry on oeis.org

0, 0, 2, 8, 15, 30, 43, 67, 90, 123, 149, 203, 237, 290, 343, 415, 464, 556, 613, 716, 800, 899, 972, 1126, 1218, 1342, 1458, 1616, 1716, 1916, 2026, 2215, 2365, 2540, 2690, 2959, 3098, 3300, 3485, 3762, 3919, 4221, 4388, 4667, 4921, 5179, 5364, 5762

Offset: 1

Benoit Cloitre and Paul D. Hanna, Nov 10 2002

nonn

These terms seem to be asymptotic to c*n^2*log(n) with c=0.69...

			These are the coefficients of x^2 in the Moebius polynomials, which begin: M(1,x)=1; M(2,x)=1 + 2x; M(3,x)=1 + 4x + 2x^2; M(4,x)=1 + 7x + 8x^2 + 2x^3; M(5,x)=1 + 9x +15x^2 +10x^3 + 2x^4; M(6,x)=1 +13x +30x^2 +27x^3 +12x^4 + 2x^5; M(7,x)=1 +15x +43x^2 +57x^3 +39x^4 +14x^5 + 2x^6; M(8,x)=1 +19x +67x^2+108x^3 +98x^4 +53x^5 +16x^6 + 2x^7.

Cf. A074586, A074587, A077596, A077597, A077599, A077600, A077601.

m[n_, 1] = 1; m[n_, k_] := m[n, k] = Sum[Floor[n/j]*m[j, k - 1], {j, 1, n - 1}]; a[n_] := m[n, 3]; Table[a[n], {n, 1, 48}] (* Jean-François Alcover, Jun 18 2013 *)

A077599 Sequence of n such that -1 is a "double" root for M(n,x) (i.e., M(n,x)=(x+1)^2*Q(n,x)).

Original entry on oeis.org

8, 9, 24, 45, 100, 117, 120, 125, 135, 171, 175, 180, 184, 224, 243, 248, 256, 261, 270, 304, 312, 324, 342, 343, 344, 360, 369, 405, 459, 468, 472, 475, 477, 486, 507, 513, 520, 531, 536, 578, 584, 603, 608, 625, 639, 640, 657, 664, 675, 704, 711, 720, 728

Offset: 1

Benoit Cloitre and Paul D. Hanna, Nov 10 2002

nonn

The n-th Moebius polynomial M(n,x) satisfies M(n,-1)=mu(n), the Moebius function of n; thus -1 is a simple root of M(n,x) if n is not squarefree. Hence these values could be called "double nonsquarefree numbers".

The n-th polynomial is divisible by (x+1)^3 for n=175, 343, 513, 800, 875. - T. D. Noe, Jan 09 2008

Cf. A074586, A074587, A077596, A077597, A077598, A077600, A077601.

a[n_,1]=1; a[n_,k_]:=a[n, k]=Sum[Floor[n/m] a[m,k-1], {m,n-1}]; t={}; Do[p=Table[a[n,k], {k,n}].(x^Range[0,n-1]); If[PolynomialMod[p,(x+1)^2]==0, AppendTo[t,n]], {n,100}]; t (* T. D. Noe, Jan 09 2008 *)

More terms from T. D. Noe, Jan 09 2008

A077600 Number of real roots for the n-th Moebius polynomial, M(n,x), which satisfies M(n,-1)=mu(n) the Moebius function of n.

Original entry on oeis.org

0, 1, 2, 3, 2, 3, 2, 5, 4, 3, 2, 5, 4, 3, 4, 5, 4, 5, 2, 5, 4, 3, 2, 7, 6, 5, 6, 5, 4, 5, 4, 7, 4, 5, 4, 7, 6, 5, 6, 7, 4, 5, 4, 5, 6, 5, 4, 7, 6, 7, 6, 5, 4, 7, 4, 5, 4, 5, 4, 7, 6, 5, 8, 7, 6, 5, 6, 9, 6, 5, 4, 9, 8, 5, 8, 9, 4, 5, 4, 7, 8, 7, 4, 7, 6, 7, 6, 7, 4, 9, 4, 7, 8, 5, 6, 7, 6, 9, 8

Offset: 1

Benoit Cloitre and Paul D. Hanna, Nov 10 2002

nonn

Cf. A074586, A074587, A077596, A077597, A077598, A077599, A077601.

m[1, x_] = 1; m[n_, x_] :=  m[n, x] = 1 + Sum[x*m[k, x]*Floor[n/k], {k, 1, n-1}] // Expand; a[n_] := CountRoots[m[n, x], x]; Table[a[n], {n, 1, 99}] (* Jean-François Alcover, Sep 13 2012 *)

Typo (?) a(45)=4 replaced with 6 by Jean-François Alcover, Sep 13 2012

A355538 Partial sum of A001221 (number of distinct prime factors) minus 1, ranging from 2 to n.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 1, 1, 2, 2, 3, 3, 4, 5, 5, 5, 6, 6, 7, 8, 9, 9, 10, 10, 11, 11, 12, 12, 14, 14, 14, 15, 16, 17, 18, 18, 19, 20, 21, 21, 23, 23, 24, 25, 26, 26, 27, 27, 28, 29, 30, 30, 31, 32, 33, 34, 35, 35, 37, 37, 38, 39, 39, 40, 42, 42, 43, 44, 46, 46

Offset: 1

Gus Wiseman, Jul 23 2022

nonn

For initial terms up to 30 we have a(n) = Log_2 A355537(n).

The sum of the same range is A000096.
The product of the same range is A000142, Heinz number A070826.
For divisors (not just prime factors) we get A002541, also A006218, A077597.
A shifted variation is A013939.
The unshifted version is A022559, product A327486, w/o multiplicity A355537.
The ranges themselves are the rows of A131818 (shifted).
Partial sums of A297155 (shifted).
A001221 counts distinct prime factors, with sum A001414.
A001222 counts prime factors with multiplicity.
A003963 multiplies together the prime indices of n.
A056239 adds up prime indices, row sums of A112798.
A066843 gives partial sums of A000005.
Cf. A000720, A002110, A076610, A305054, A355538, A355731, A355733, A355741, A355744, A355745, A355746, A355747.

Table[Total[(PrimeNu[#]-1)&/@Range[2,n]],{n,1,100}]

a(n) = A013939(n) - n + 1.

A085683 a(n) = Sum_{k = 1..N-1} floor(N/k) where N is the n-th prime.

Original entry on oeis.org

2, 4, 9, 15, 28, 36, 51, 59, 75, 102, 112, 141, 159, 169, 187, 218, 248, 262, 293, 313, 327, 357, 378, 412, 460, 483, 493, 515, 529, 553, 636, 658, 696, 706, 767, 781, 821, 857, 877, 918, 952, 972, 1032, 1048, 1071, 1085, 1167, 1239, 1266, 1280, 1306, 1342, 1364, 1422

Offset: 1

N. J. A. Sloane, Oct 28 2008

nonn

The old entry with this sequence number was a duplicate of A081532.

Giovanni Resta, Table of n, a(n) for n = 1..10000
R. K. Guy, Conway's prime producing machine, Math. Mag. 56 (1983), no. 1, 26-33 (see p. 33).

Cf. A006218, A077597.

(Rest@ FoldList[ Plus, 0, DivisorSigma[0, Range@ Prime@ 100]])[[ Prime@ Range@ 100]] -1 (* Giovanni Resta, Jun 09 2015 *)

from math import isqrt
from sympy import prime
def A085683(n): return -(s:=isqrt(m:=prime(n)))**2+(sum(m//k for k in range(1,s+1))<<1)-1 # Chai Wah Wu, Oct 23 2023

cp's OEIS Frontend

A077596 Central coefficients of Moebius polynomials (A074586): coefficient of x^(n/2-1/2) if n is odd; coefficient of x^(n/2-1) if n is even and >4. The n-th Moebius polynomial, M(n,x), satisfies M(n,-1)=mu(n) the Moebius function of n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A077598 Coefficient of x^2 in the n-th Moebius polynomial (A074586), M(n,x), which satisfies M(n,-1)=mu(n) the Moebius function of n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A077599 Sequence of n such that -1 is a "double" root for M(n,x) (i.e., M(n,x)=(x+1)^2*Q(n,x)).

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Extensions

A077600 Number of real roots for the n-th Moebius polynomial, M(n,x), which satisfies M(n,-1)=mu(n) the Moebius function of n.

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Extensions

A355538 Partial sum of A001221 (number of distinct prime factors) minus 1, ranging from 2 to n.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Formula

A085683 a(n) = Sum_{k = 1..N-1} floor(N/k) where N is the n-th prime.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Python