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 1-4 of 4 results.

A064945 a(n) = Sum_{i|n, j|n, j >= i} i.

Original entry on oeis.org

1, 4, 5, 11, 7, 22, 9, 26, 18, 30, 13, 64, 15, 38, 38, 57, 19, 82, 21, 87, 48, 54, 25, 156, 38, 62, 58, 109, 31, 179, 33, 120, 68, 78, 68, 244, 39, 86, 78, 213, 43, 224, 45, 153, 143, 102, 49, 348, 66, 166, 98, 175, 55, 268, 96, 267, 108, 126, 61, 542, 63, 134, 181
Offset: 1

Views

Author

Vladeta Jovovic, Oct 28 2001

Keywords

Examples

			a(6) = dot_product(4,3,2,1)*(1,2,3,6) = 4*1+3*2+2*3+1*6 = 22.

Links

Crossrefs

Cf. A060640, A064944, A064946, A064947, A064948, A064949.
Cf. A027750, A135539, A000005, A000203, A064840, A337360, A337297.
Cf. A386911.

Programs

  • Haskell
    a064945 = sum . zipWith (*) [1..] . reverse . a027750_row'
-- Reinhard Zumkeller, Jul 14 2015
  • Maple
    with(numtheory): seq(add((tau(n)-i+1)*sort(convert(divisors(n),'list'))[i],i=1..tau(n)), n=1..200);
  • Mathematica
    A064945[n_] := #.Range[Length[#], 1, -1] & [Divisors[n]];
Array[A064945, 100] (* Paolo Xausa, Aug 07 2025 *)
  • PARI
    a(n) = my(d=divisors(n), t=length(d)); sum(i=1, t, (t - i + 1)*d[i]); \\ Harry J. Smith, Oct 01 2009
  • PARI
    a(n)=my(d=divisors(n)); sum(i=1,#d,(#d+1-i)*d[i]) \\ Charles R Greathouse IV, Jun 10 2015
  • Python
    from sympy import divisors, divisor_sigma
def A064945(n): return (divisor_sigma(n,0)+1)*divisor_sigma(n)-sum(a*b for a, b in enumerate(divisors(n),1)) # Chai Wah Wu, Aug 07 2025

Formula

a(n) = Sum_{i=1..tau(n)} (tau(n)-i+1)*d_i, where {d_i}, i=1..tau(n), is the increasing sequence of divisors of n.
a(n) = Sum_{i=1..A000005(n)} (A000005(n)-i+1)*A027750(n, i). - Michel Marcus, Jun 10 2015
From Ridouane Oudra, Aug 02 2025: (Start)
a(n) = Sum_{d|n} d*A135539(n,d).
a(n) = A064947(n) + A000203(n).
a(n) = (A064949(n) + A000203(n))/2.
a(n) = A064949(n) - A064947(n).
a(n) = A337360(n) - A064944(n).
a(n) = A064840(n) - A064946(n). (End)

A064944 a(n) = Sum_{i|n, j|n, j >= i} j.

Original entry on oeis.org

1, 5, 7, 17, 11, 38, 15, 49, 34, 60, 23, 132, 27, 82, 82, 129, 35, 191, 39, 207, 112, 126, 47, 384, 86, 148, 142, 283, 59, 469, 63, 321, 172, 192, 172, 666, 75, 214, 202, 597, 83, 640, 87, 435, 403, 258, 95, 1016, 162, 485, 262, 511, 107, 812, 264, 813, 292, 324
Offset: 1

Views

Author

Vladeta Jovovic, Oct 28 2001

Keywords

Examples

			a(6) = max(1,1)+max(1,2)+max(1,3)+max(1,6)+max(2,2)+max(2,3)+max(2,6)+max(3,3)+max(3,6)+max(6,6)=38, or a(6) = dot_product(1,2,3,4)*(1,2,3,6)=1*1+2*2+3*3+4*6=38.

Links

Crossrefs

Cf. A060640, A064945, A064946, A064947, A064948, A064949.
Cf. A027750, A135539, A000005, A000203, A064840, A337360, A337297.

Programs

  • Haskell
    a064944 = sum . zipWith (*) [1..] . a027750_row'
-- Reinhard Zumkeller, Jul 14 2015
  • Maple
    with(numtheory): seq(add(i*sort(convert(divisors(n),'list'))[i],i=1..tau(n)), n=1..200);
  • Mathematica
    A064944[n_] := #.Range[Length[#]] & [Divisors[n]];
Array[A064944, 100] (* Paolo Xausa, Aug 07 2025 *)
  • PARI
    a(n) = my(d=divisors(n)); sum(i=1, length(d), i*d[i]); \\ Harry J. Smith, Sep 30 2009
  • Python
    from sympy import divisors
def A064944(n): return sum(a*b for a, b in enumerate(divisors(n),1)) # Chai Wah Wu, Aug 07 2025

Formula

a(n) = Sum_{i=1..tau(n)} i*d_i, where {d_i}, i=1..tau(n) is the increasing sequence of divisors of n.
a(n) = Sum_{i=1..A000005(n)} i*A027750(n, i). - Michel Marcus, Jun 10 2015
From Ridouane Oudra, Aug 01 2025: (Start)
a(n) = Sum_{d|n} (n/d)*A135539(n,d).
a(n) = A064946(n) + A000203(n).
a(n) = (A064948(n) + A000203(n))/2.
a(n) = A337360(n) - A064945(n).
a(n) = A064948(n) - A064946(n).
a(n) = A064840(n) - A064947(n). (End)

A064946 a(n) = Sum_{i|n, j|n, j>i} j.

Original entry on oeis.org

0, 2, 3, 10, 5, 26, 7, 34, 21, 42, 11, 104, 13, 58, 58, 98, 17, 152, 19, 165, 80, 90, 23, 324, 55, 106, 102, 227, 29, 397, 31, 258, 124, 138, 124, 575, 37, 154, 146, 507, 41, 544, 43, 351, 325, 186, 47, 892, 105, 392, 190, 413, 53, 692, 192, 693, 212, 234, 59
Offset: 1

Views

Author

Vladeta Jovovic, Oct 28 2001

Keywords

Examples

			a(6) = dot_product(0,1,2,3)*(1,2,3,6) = 0*1 + 1*2 + 2*3 + 3*6 = 26.

Links

Crossrefs

Cf. A064944, A064945, A064947, A064948, A064949.
Cf. A000005, A000203, A064840, A337297, A027750.

Programs

  • Maple
    with(numtheory): seq(add((i-1)*sort(convert(divisors(n),'list'))[i],i=1..tau(n)), n=1..200);
  • Mathematica
    A064946[n_] := #.Range[Length[#]] & [Rest[Divisors[n]]];
Array[A064946, 100] (* Paolo Xausa, Aug 14 2025 *)
  • PARI
    a(n) = my(d=divisors(n)); sum(i=2, length(d), (i - 1)*d[i]); \\ Harry J. Smith, Oct 01 2009

Formula

a(n) = Sum_{i=1..tau(n)} (i-1)*d_i, where {d_i}, i=1..tau(n), is the increasing sequence of the divisors of n.
a(n) = A064944(n) - A000203(n). - Amiram Eldar, Dec 23 2024
From Ridouane Oudra, Aug 06 2025: (Start)
a(n) = A064948(n) - A064944(n).
a(n) = A064840(n) - A064945(n).
a(n) = A337297(n) - A064947(n).
a(n) = (A064948(n) - A000203(n))/2. (End)

A064947 a(n) = Sum_{i|n, j|n, j>i} i.

Original entry on oeis.org

0, 1, 1, 4, 1, 10, 1, 11, 5, 12, 1, 36, 1, 14, 14, 26, 1, 43, 1, 45, 16, 18, 1, 96, 7, 20, 18, 53, 1, 107, 1, 57, 20, 24, 20, 153, 1, 26, 22, 123, 1, 128, 1, 69, 65, 30, 1, 224, 9, 73, 26, 77, 1, 148, 24, 147, 28, 36, 1, 374, 1, 38, 77, 120, 26, 168, 1, 93, 32, 165, 1, 411, 1, 44
Offset: 1

Views

Author

Vladeta Jovovic, Oct 28 2001

Keywords

Comments

For given n, iterate a(n); a(a(n)); a(a(a(n))); ... Does this iterative process always lead to a(a(...(a(n))...)) = 1? - Ctibor O. Zizka, Apr 17 2008
No. For example, a(4) = 4, a(14) = 14, and a(99) = 99. - Jason Yuen, Jan 07 2025

Examples

			a(6) = dot_product(3,2,1,0)*(1,2,3,6) = 3*1 + 2*2 + 1*3 + 0*6 = 10.

Links

Crossrefs

Cf. A000005, A000203, A064840, A337297, A027750.

Programs

  • Maple
    with(numtheory): seq(add((tau(n)-i)*sort(convert(divisors(n),'list'))[i],i=1..tau(n)), n=1..200);
  • Mathematica
    Table[t = DivisorSigma[0, n]; Total@ MapIndexed[(t - First[#2])*#1 &, Divisors[n]], {n, 120}] (* Michael De Vlieger, Jan 07 2025 *)
  • PARI
    a(n) = my(d=divisors(n), t=length(d)); sum(i=1, t - 1, (t - i)*d[i]); \\ Harry J. Smith, Oct 01 2009

Formula

a(n) = Sum_{i=1..tau(n)} (tau(n)-i)*d_i, where {d_i}, i=1..tau(n), is increasing sequence of divisors of n.
From Ridouane Oudra, Aug 07 2025: (Start)
a(n) = A064945(n) - A000203(n).
a(n) = A064840(n) - A064944(n).
a(n) = A064949(n) - A064945(n).
a(n) = A337297(n) - A064946(n).
a(n) = (A064949(n) - A000203(n))/2. (End)
Showing 1-4 of 4 results.

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