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.

User: Donald S. McDonald

Donald S. McDonald's wiki page.

Donald S. McDonald has authored 54 sequences. Here are the ten most recent ones:

A340959 Table read by antidiagonals of the smallest prime >= n^k, n >= 1 and k >= 0.

Original entry on oeis.org

2, 2, 2, 2, 2, 2, 2, 3, 5, 2, 2, 5, 11, 11, 2, 2, 5, 17, 29, 17, 2, 2, 7, 29, 67, 83, 37, 2, 2, 7, 37, 127, 257, 251, 67, 2, 2, 11, 53, 223, 631, 1031, 733, 131, 2, 2, 11, 67, 347, 1297, 3137, 4099, 2203, 257, 2, 2, 11, 83, 521, 2411, 7789, 15629, 16411, 6563
Offset: 1

Views

Author

Donald S. McDonald, Jan 31 2021

Keywords

Examples

			Table begins:
  2, 2,  2,   2,   2,    2, ...
  2, 2,  5,  11,  17,   37, ...
  2, 3, 11,  29,  83,  251, ...
  2, 5, 17,  67, 257, 1031, ...
  2, 5, 29, 127, 631, 3137, ...
  ...;
yielding the triangle:
  2;
  2, 2;
  2, 2,  2;
  2, 3,  5,  2;
  2, 5, 11, 11,  2;
  2, 5, 17, 29, 17, 2;
  ...

Crossrefs

Cf. A104080 (n=2), A104081 (n=3), A104082 (n=4), A104083 (n=5), A104084 (n=7).

Programs

  • Mathematica
    T[n_,k_]:=NextPrime[n^k-1];Flatten[Table[T[n-k,k],{n,11},{k,0,n-1}]] (* Stefano Spezia, Feb 01 2021 *)
  • PARI
    T(n,k) = nextprime(n^k); \\ Michel Marcus, Feb 01 2021

Formula

T(n,k) = next_prime(n^k-1).

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)

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

A328817 Numbers k such that at least 7 of k, k+1, ..., k+9 are divisible by their least prime factor squared.

Original entry on oeis.org

3475, 18271, 25524, 25623, 45616, 55772, 72471, 72472, 104419, 121667, 133223, 133224, 149220, 164975, 165568, 165571, 172916, 180167, 180168, 203979, 203980, 219123, 260424, 261472, 261475, 334516, 334519, 364216, 381267, 393320, 393323, 402723, 412524, 420467, 420468
Offset: 1

Views

Author

Sean A. Irvine (on behalf of Donald S. McDonald), Nov 07 2019

Keywords

Examples

			203980 is a member since 2^2 | 203980, 37^2 | 203981, 13^2 | 203983, 2^2 | 203984, 3^2 | 203985, 7^2 | 203987, and 2^2 | 203988.

Links

Crossrefs

Cf. A283050.

Programs

  • Mathematica
    Flatten[Position[Partition[Table[If[Divisible[n,FactorInteger[n][[1,1]]^2],1,0],{n,420000}],10,1],?(Total[#]>6&)]] (* _Harvey P. Dale, Jan 02 2021 *)
  • PARI
    upto(n) = {my(l, c, res); l = List([0,0,0,1,0,0,0,1,1,0]); c = 3; res = List(); for(i = 11, n, f = factor(i)[,2]; c -= l[1]; listpop(l, 1); if(f[1] >= 2, c++; listput(l, 1) , listput(l, 0) ); if(c >= 7, listput(res, i-9); ) ); res } \\ David A. Corneth, Jan 02 2021

A316850 The table in A316842 with columns concatenated to form a single number.

Original entry on oeis.org

111, 221, 322, 331, 332, 432, 433, 441, 443, 533, 542, 543, 544, 551, 552, 553, 554, 643, 652, 653, 654, 655, 661, 665, 744, 753, 754, 755, 762, 763, 764, 765, 766, 771, 772, 773, 774, 775, 776, 854, 855, 863, 865, 872, 873, 874, 875, 876, 877, 881, 883, 885, 887
Offset: 1

Views

Author

N. J. A. Sloane, Jul 23 2018, following a suggestion from Donald S. McDonald

Keywords

Comments

This makes no sense once the entries in A316842 exceed 9, but is included because some people may search for this version. See A316842 for the official version.
Does NOT need a b-file.

Crossrefs

Cf. A316842.

A316849 The table in A316841 with columns concatenated to form a single number.

Original entry on oeis.org

111, 221, 222, 322, 331, 332, 333, 432, 433, 441, 442, 443, 444, 533, 542, 543, 544, 551, 552, 553, 554, 555, 643, 644, 652, 653, 654, 655, 661, 662, 663, 664, 665, 666, 744, 753, 754, 755, 762, 763, 764, 765, 766, 771, 772, 773, 774, 775, 776, 777, 854, 855
Offset: 1

Views

Author

N. J. A. Sloane, Jul 23 2018, following a suggestion from Donald S. McDonald

Keywords

Comments

This makes no sense once the entries in A316841 exceed 9, but is included because some people may search for this version. See A316841 for the official version.
Does NOT need a b-file.

Crossrefs

Cf. A316841.

A316842 Three-column table read by rows giving primitive integer sides of proper triangles (i,j,k) with i >= j >= k >= 1, j+k > i, gcd(i,j,k) = 1.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 3, 2, 2, 3, 3, 1, 3, 3, 2, 4, 3, 2, 4, 3, 3, 4, 4, 1, 4, 4, 3, 5, 3, 3, 5, 4, 2, 5, 4, 3, 5, 4, 4, 5, 5, 1, 5, 5, 2, 5, 5, 3, 5, 5, 4, 6, 4, 3, 6, 5, 2, 6, 5, 3, 6, 5, 4, 6, 5, 5, 6, 6, 1, 6, 6, 5, 7, 4, 4, 7, 5, 3, 7, 5, 4, 7, 5, 5, 7, 6, 2, 7, 6, 3, 7, 6, 4, 7, 6, 5, 7, 6, 6, 7, 7, 1, 7, 7, 2, 7, 7, 3, 7, 7, 4, 7, 7, 5, 7, 7, 6, 8, 5, 4
Offset: 1

Views

Author

N. J. A. Sloane, Jul 23 2018, following a suggestion from Donald S. McDonald

Keywords

Examples

			Table begins:
[1,1,1],
[2,2,1],
[3,2,2],
[3,3,1],
[3,3,2],
[4,3,2],
[4,3,3],
[4,4,1],
[4,4,3],
[5,3,3],
[5,4,2],
...

Links

Crossrefs

There are A123323(k) rows that begin with k.
The three columns are A316846, A316847, A316848.
A316850 is a compressed version.
See A316841 for all triples (including imprimitive triples).
See A316852 and A317181 & A317183 for perimeter and area.
Other related sequences: A051493, A070080, A070081, A070082, A070110.

A316841 Three-column table read by rows giving integer sides of proper triangles (i,j,k) with i >= j >= k >= 1, j+k > i.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 2, 2, 2, 3, 2, 2, 3, 3, 1, 3, 3, 2, 3, 3, 3, 4, 3, 2, 4, 3, 3, 4, 4, 1, 4, 4, 2, 4, 4, 3, 4, 4, 4, 5, 3, 3, 5, 4, 2, 5, 4, 3, 5, 4, 4, 5, 5, 1, 5, 5, 2, 5, 5, 3, 5, 5, 4, 5, 5, 5, 6, 4, 3, 6, 4, 4, 6, 5, 2, 6, 5, 3, 6, 5, 4, 6, 5, 5, 6, 6, 1, 6, 6, 2, 6, 6, 3, 6, 6, 4, 6, 6, 5, 6, 6, 6, 7, 4, 4, 7, 5, 3, 7, 5, 4, 7, 5, 5, 7, 6, 2, 7, 6, 3, 7, 6, 4, 7, 6, 5
Offset: 1

Views

Author

N. J. A. Sloane, Jul 23 2018, following a suggestion from Donald S. McDonald

Keywords

Examples

			Table begins (imprimitive triples are labeled i):
[1,1,1],
[2,2,1],
[2,2,2],i
[3,2,2],
[3,3,1],
[3,3,2],
[3,3,3],i
[4,3,2],
[4,3,3],
[4,4,1],
[4,4,2],i
[4,4,3],
[4,4,4],i
[5,3,3],
...

Links

Crossrefs

There are A002620(k+1) rows that begin with k.
The three columns are A316843, A316844, A316845.
A316849 is a compressed version.
See A316842 for primitive triples.
See A316851 and A316853 & A317182 for perimeter and area.
Other related sequences: A051493, A070080, A070081, A070082, A070110.

Programs

  • PARI
    for(i=1,6, for(j=1,i, for(k=1,j, if(j+k>i, print1(i,", ",j,", ",k,", "))))) \\ Hugo Pfoertner, Jan 25 2020

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.