cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

A336276 a(n) = Sum_{k=1..n} mu(k)*k^2.

Original entry on oeis.org

1, -3, -12, -12, -37, -1, -50, -50, -50, 50, -71, -71, -240, -44, 181, 181, -108, -108, -469, -469, -28, 456, -73, -73, -73, 603, 603, 603, -238, -1138, -2099, -2099, -1010, 146, 1371, 1371, 2, 1446, 2967, 2967, 1286, -478, -2327, -2327, -2327, -211, -2420
Offset: 1

Views

Author

Donald S. McDonald, Jul 15 2020

Keywords

Comments

Conjecture: a(n) changes sign infinitely often.

Links

Crossrefs

Cf. A002321, A068340, A336277, A336278, A336279.
Cf. A008683, A055615, A070891, A360390, A361983.

Programs

  • Mathematica
    Array[Sum[MoebiusMu[k]*k^2, {k, #}] &, 47] (* Michael De Vlieger, Jul 15 2020 *)
  • PARI
    a(n) = sum(k=1, n, moebius(k)*k^2); \\ Michel Marcus, Jul 15 2020
  • Python
    from functools import lru_cache
@lru_cache(maxsize=None)
def A336276(n):
    if n <= 1:
        return 1
    c, j = 1, 2
    k1 = n//j
    while k1 > 1:
        j2 = n//k1 + 1
        c -= (j2*(j2-1)*((j2<<1)-1)-j*(j-1)*((j<<1)-1))//6*A336276(k1)
        j, k1 = j2, n//j2
    return c-(n*(n+1)*((n<<1)+1)-j*(j-1)*((j<<1)-1))//6 # Chai Wah Wu, Apr 04 2023

Formula

Partial sums of A334657.
G.f. A(x) satisfies x = Sum_{k>=1} k^2 * (1 - x^k) * A(x^k). - Seiichi Manyama, Apr 01 2023
Sum_{k=1..n} k^2 * a(floor(n/k)) = 1. - Seiichi Manyama, Apr 03 2023

A336277 a(n) = Sum_{k=1..n} mu(k)*k^3.

Original entry on oeis.org

1, -7, -34, -34, -159, 57, -286, -286, -286, 714, -617, -617, -2814, -70, 3305, 3305, -1608, -1608, -8467, -8467, 794, 11442, -725, -725, -725, 16851, 16851, 16851, -7538, -34538, -64329, -64329, -28392, 10912, 53787, 53787, 3134, 58006, 117325, 117325, 48404
Offset: 1

Views

Author

Donald S. McDonald, Jul 15 2020

Keywords

Comments

Conjecture: a(n) changes sign infinitely often.

Links

Crossrefs

Cf. A002321, A068340, A336276, A336278, A336279.
Cf. A008683, A055615, A070891.

Programs

  • Mathematica
    Array[Sum[MoebiusMu[k]*k^3, {k, #}] &, 41] (* Michael De Vlieger, Jul 15 2020 *)
Accumulate[Table[MoebiusMu[n] n^3,{n,50}]] (* Harvey P. Dale, Aug 15 2024 *)
  • PARI
    a(n) = sum(k=1, n, moebius(k)*k^3); \\ Michel Marcus, Jul 15 2020
  • Python
    from functools import lru_cache
@lru_cache(maxsize=None)
def A336277(n):
    if n <= 1:
        return 1
    c, j = 1, 2
    k1 = n//j
    while k1 > 1:
        j2 = n//k1 + 1
        c -= ((j2*(j2-1))**2-(j*(j-1))**2>>2)*A336277(k1)
        j, k1 = j2, n//j2
    return c-((n*(n+1))**2-((j-1)*j)**2>>2) # Chai Wah Wu, Apr 04 2023

Formula

Partial sums of A334659.
G.f. A(x) satisfies x = Sum_{k>=1} k^3 * (1 - x^k) * A(x^k). - Seiichi Manyama, Apr 01 2023
Sum_{k=1..n} k^3 * a(floor(n/k)) = 1. - Seiichi Manyama, Apr 03 2023

A336278 a(n) = Sum_{k=1..n} mu(k)*k^4.

Original entry on oeis.org

1, -15, -96, -96, -721, 575, -1826, -1826, -1826, 8174, -6467, -6467, -35028, 3388, 54013, 54013, -29508, -29508, -159829, -159829, 34652, 268908, -10933, -10933, -10933, 446043, 446043, 446043, -261238, -1071238, -1994759, -1994759, -808838, 527498, 2028123
Offset: 1

Views

Author

Donald S. McDonald, Jul 15 2020

Keywords

Comments

Conjecture: a(n) changes sign infinitely often.

Links

Crossrefs

Cf. A002321, A068340, A336276, A336277, A336279.
Cf. A008683, A055615, A070891, A344429, A344430.

Programs

  • Mathematica
    Array[Sum[MoebiusMu[k]*k^4, {k, #}] &, 35] (* Michael De Vlieger, Jul 15 2020 *)
Accumulate[Table[MoebiusMu[x]x^4,{x,40}]] (* Harvey P. Dale, Jan 14 2021 *)
  • PARI
    a(n) = sum(k=1, n, moebius(k)*k^4); \\ Michel Marcus, Jul 15 2020
  • Python
    from functools import lru_cache
@lru_cache(maxsize=None)
def A336278(n):
    if n <= 1:
        return 1
    c, j = 1, 2
    k1 = n//j
    while k1 > 1:
        j2 = n//k1 + 1
        c -= (j2*(j2**2*(j2*(6*j2 - 15) + 10) - 1)-j*(j**2*(j*(6*j - 15) + 10) - 1))//30*A336278(k1)
        j, k1 = j2, n//j2
    return c-(n*(n**2*(n*(6*n + 15) + 10) - 1)-j*(j**2*(j*(6*j - 15) + 10) - 1))//30 # Chai Wah Wu, Apr 04 2023

Formula

Partial sums of A334660.
From Seiichi Manyama, Apr 03 2023: (Start)
G.f. A(x) satisfies x = Sum_{k>=1} k^4 * (1 - x^k) * A(x^k).
Sum_{k=1..n} k^4 * a(floor(n/k)) = 1. (End)

A336279 a(n) = Sum_{k=1..n} mu(k)*k^5.

Original entry on oeis.org

1, -31, -274, -274, -3399, 4377, -12430, -12430, -12430, 87570, -73481, -73481, -444774, 93050, 852425, 852425, -567432, -567432, -3043531, -3043531, 1040570, 6194202, -242141, -242141, -242141, 11639235, 11639235, 11639235, -8871914, -33171914, -61801065
Offset: 1

Views

Author

Donald S. McDonald, Jul 15 2020

Keywords

Comments

Conjecture: a(n) changes sign infinitely often.

Links

Crossrefs

Cf. A002321, A068340, A336276, A336277, A336278.
Cf. A008683, A055615, A070891, A344429, A344430.

Programs

  • Mathematica
    Array[Sum[MoebiusMu[k]*k^5, {k, #}] &, 32] (* Michael De Vlieger, Jul 15 2020 *)
  • PARI
    a(n) = sum(k=1, n, moebius(k)*k^5); \\ Michel Marcus, Jul 15 2020
  • Python
    from functools import lru_cache
@lru_cache(maxsize=None)
def A336279(n):
    if n <= 1:
        return 1
    c, j = 1, 2
    k1 = n//j
    while k1 > 1:
        j2 = n//k1 + 1
        c -= (j2**2*(j2**2*(j2*(2*j2 - 6) + 5) - 1)-j**2*(j**2*(j*(2*j - 6) + 5) - 1))//12*A336279(k1)
        j, k1 = j2, n//j2
    return c-(n**2*(n**2*(n*(2*n + 6) + 5) - 1)-j**2*(j**2*(j*(2*j - 6) + 5) - 1))//12 # Chai Wah Wu, Apr 04 2023

Formula

From Seiichi Manyama, Apr 03 2023: (Start)
G.f. A(x) satisfies x = Sum_{k>=1} k^5 * (1 - x^k) * A(x^k).
Sum_{k=1..n} k^5 * a(floor(n/k)) = 1. (End)

A070890 Numerator of Sum_{k=1..n} mu(k)/k when it changes sign.

Original entry on oeis.org

-1, 2, -1, 19, -1, 304, -81988, 410857, -249979, 4165258, -65721449, 2562470143, -5468849774, 184344882947, -137190436674212, 10026981687881, -12611493192339623, 519973962150962777, -8549627883788520181, 1874648830674470878723, -200643437220052588790575, 877316785444551755504875
Offset: 1

Views

Author

Donald S. McDonald, May 17 2002

Keywords

Examples

			a(1)= -1 because numerator first sign change is 1-1/2-1/3-1/5= > (-1)/30 <0.
a(2)= 2 because next sign change is -1/30+1/6 = 2/15, reverts to positive.

Crossrefs

Cf. A070891, A070888, A070889.

Programs

  • PARI
    t = 0; v = []; for( n = 1, 120, t1 = t; t = t + moebius(n) / n; if( t * t1 < 0, v = concat( v, numerator( t)), )); v

A070892 Numbers n such that absolute value of Sum_{k=1..n} mu(k)/k sets a new minimum.

Original entry on oeis.org

1, 2, 3, 5, 7, 11, 61, 154, 857, 859, 2141, 2153, 2161, 39011, 39065, 39095, 56026, 56045, 56101, 56189, 56242, 56245, 56254, 56263, 56359, 2985634, 2985703, 2986715, 2986718, 2986721, 16904494, 16904497, 16904531, 16904534
Offset: 1

Views

Author

Donald S. McDonald, May 17 2002

Keywords

Crossrefs

Cf. A070888-A070891.

Programs

  • Mathematica
    s = 1; t = 0; Do[t = t + MoebiusMu[n] / n; If[ s > Abs[t], s = Abs[t]; Print[n]], {n, 1, 2 * 10^7}]
  • PARI
    t = 0.; t1 = 1; v = []; for( n = 1, 1400, t = t + moebius( n) / n; if( (t / t1 )^2 < 1, t1 = t; v = concat( v, n), )); v

Extensions

Edited and extended by Robert G. Wilson v, May 24 2002
Showing 1-6 of 6 results.

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