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.

A022825 a(n) = a([ n/2 ]) + a([ n/3 ]) + . . . + a([ n/n ]) for n > 1, a(1) = 1.

Original entry on oeis.org

1, 1, 2, 3, 4, 6, 7, 9, 11, 13, 14, 19, 20, 22, 25, 29, 30, 36, 37, 42, 45, 47, 48, 60, 62, 64, 68, 73, 74, 84, 85, 93, 96, 98, 101, 119, 120, 122, 125, 137, 138, 148, 149, 154, 162, 164, 165, 193, 195, 201, 204, 209, 210, 226, 229, 241, 244, 246, 247, 278, 279
Offset: 1

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A025523, A078346, A345182.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n<2, 1,
      add(a(iquo(n,j)), j=2..n))
    end:
seq(a(n), n=1..63);  # Alois P. Heinz, Mar 31 2021
  • Mathematica
    Fold[Append[#1, Total[#1[[Quotient[#2, Range[2, #2]]]]]] &, {1}, Range[2, 60]] (* Ivan Neretin, Aug 24 2016 *)
  • Python
    from functools import lru_cache
@lru_cache(maxsize=None)
def A022825(n):
    if n <= 1:
        return n
    c, j = 0, 2
    k1 = n//j
    while k1 > 1:
        j2 = n//k1 + 1
        c += (j2-j)*A022825(k1)
        j, k1 = j2, n//j2
    return c+n+1-j # Chai Wah Wu, Mar 31 2021

Formula

G.f. A(x) satisfies: A(x) = x + (1/(1 - x)) * Sum_{k>=2} (1 - x^k) * A(x^k). - Ilya Gutkovskiy, Feb 21 2022

Extensions

Offset corrected by Alois P. Heinz, Mar 31 2021

A097417 a(1)=1; a(n+1) = Sum_{k=1..n} a(k) a(floor(n/k)).

Original entry on oeis.org

1, 1, 2, 5, 13, 34, 90, 236, 621, 1629, 4274, 11193, 29337, 76818, 201173, 526730, 1379178, 3610804, 9453695, 24750281, 64798235, 169644626, 444138288, 1162770238, 3044180080, 7969770106, 20865148382, 54625676431, 143011928942
Offset: 1

Views

Author

Leroy Quet, Aug 19 2004

Keywords

Comments

4 is the only composite number n such that a(n+1) = 3a(n) - a(n-1) and if n is a composite number greater than 4 then a(n+1) > 3a(n) - a(n-1). - Farideh Firoozbakht, Feb 05 2005

Links

Crossrefs

Cf. A038044, A078346, A097919.

Programs

  • Maple
    a[1]:=1: for n from 1 to 50 do: a[n+1]:=sum(a[k]*a[floor(n/k)],k=1..n): od: seq(a[i],i=1..51) # Mark Hudson, Aug 21 2004
  • Mathematica
    a[1] = 1; a[n_] := a[n] = Sum[ a[k]*a[Floor[(n - 1)/k]], {k, n - 1}]; Table[ a[n], {n, 29}] (* Robert G. Wilson v, Aug 21 2004 *)
  • PARI
    {m=29;a=vector(m);print1(a[1]=1,",");for(n=1,m-1,print1(a[n+1]=sum(k=1,n,a[k]*a[floor(n/k)]),","))} \\ Klaus Brockhaus, Aug 21 2004

Formula

Ratio a(n+1)/a(n) seems to tend to 1 + Golden Ratio = 2.61803398... = 1 + A001622. - Mark Hudson (mrmarkhudson(AT)hotmail.com), Aug 23 2004
Satisfies the "partial linear recursion": a(prime(n)+1) = 3*a(prime(n)) - a(prime(n)-1). This explains why we get a(n+1)/a(n) -> 1 + phi. Also, lim_{n->oo} a(n)/(1 + phi)^n exists but should not have a simple closed form. - Benoit Cloitre, Aug 29 2004
Limit_{n->oo} a(n)/(1 + phi)^n = 0.108165624886204570982244311730754895284041534583990405146651275318889227986... - Vaclav Kotesovec, May 28 2021

Extensions

More terms from Klaus Brockhaus, Robert G. Wilson v and Mark Hudson (mrmarkhudson(AT)hotmail.com), Aug 21 2004

A290845 a(1) = 1; a(n) = Sum_{k=1..n} a(ceiling((n-1)/k)).

Original entry on oeis.org

1, 2, 4, 8, 14, 24, 36, 56, 78, 110, 148, 200, 254, 334, 416, 522, 644, 798, 954, 1162, 1372, 1640, 1934, 2284, 2636, 3090, 3556, 4106, 4694, 5394, 6096, 6972, 7850, 8882, 9972, 11220, 12500, 14048, 15598, 17360, 19208, 21346, 23486, 26016, 28548, 31436, 34478, 37874, 41272, 45246
Offset: 1

Views

Author

Ilya Gutkovskiy, Aug 12 2017

Keywords

Examples

			a(1) = 1;
a(2) = a(ceiling(1/1)) + a(ceiling(1/2)) = a(1) + a(1) = 2;
a(3) = a(ceiling(2/1)) + a(ceiling(2/2)) + a(ceiling(2/3)) = a(2) + a(1) + a(1) = 4, etc.

Crossrefs

Cf. A003318, A025523, A068336 (first differences), A078346.

Programs

  • Mathematica
    a[1] = 1; a[n_] := a[n] = Sum[a[Ceiling[(n - 1)/k]], {k, 1, n}]; Table[a[n], {n, 50}]
  • Python
    from functools import lru_cache
@lru_cache(maxsize=None)
def A290845(n):
    if n == 1:
        return 1
    c, j, k1 = n, 1, n-2
    while k1 > 1:
        j2 = (n-2)//k1 + 1
        c += (j2-j)*A290845(k1+1)>>1
        j, k1 = j2, (n-2)//j2
    return c-j<<1 # Chai Wah Wu, Apr 29 2025

Formula

a(n) = 2*A003318(n-1) for n > 1.

A347028 a(1) = 1; a(n+1) = -Sum_{k=1..n} a(floor(n/k)).

Original entry on oeis.org

1, -1, 0, -2, 1, -3, 1, -4, 4, -6, 2, -7, 8, -8, 5, -13, 13, -14, 9, -15, 19, -21, 12, -22, 32, -26, 18, -36, 33, -37, 31, -38, 57, -48, 32, -56, 66, -57, 44, -74, 83, -75, 65, -76, 100, -102, 68, -103, 140, -108, 94, -136, 140, -137, 119, -149, 193, -174, 125, -175, 228, -176, 161, -224, 256
Offset: 1

Views

Author

Ilya Gutkovskiy, Aug 11 2021

Keywords

Crossrefs

Cf. A078346, A309288, A328967.

Programs

  • Mathematica
    a[1] = 1; a[n_] := a[n] = -Sum[a[Floor[(n - 1)/k]], {k, 1, n - 1}]; Table[a[n], {n, 1, 65}]
nmax = 65; A[] = 0; Do[A[x] = x - (x/(1 - x)) Sum[(1 - x^k) A[x^k], {k, 1, nmax}] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] // Rest
  • Python
    from functools import lru_cache
@lru_cache(maxsize=None)
def A347028(n):
    if n == 1:
        return 1
    c, j, k1 = n, 1, n-1
    while k1 > 1:
        j2 = (n-1)//k1 + 1
        c += (j2-j)*A347028(k1)
        j, k1 = j2, (n-1)//j2
    return j-c # Chai Wah Wu, Apr 29 2025

Formula

G.f. A(x) satisfies: A(x) = x - (x/(1 - x)) * Sum_{k>=1} (1 - x^k) * A(x^k).

A351620 a(1) = 1; a(n) = a(n-1) + Sum_{k=2..n} a(floor(n/k)).

Original entry on oeis.org

1, 2, 4, 8, 13, 22, 32, 48, 67, 93, 120, 164, 209, 266, 331, 418, 506, 626, 747, 905, 1076, 1276, 1477, 1755, 2039, 2370, 2723, 3149, 3576, 4112, 4649, 5295, 5971, 6737, 7519, 8501, 9484, 10590, 11744, 13109, 14475, 16092, 17710, 19561, 21504, 23650, 25797, 28386, 30986, 33903
Offset: 1

Views

Author

Ilya Gutkovskiy, Feb 20 2022

Keywords

Comments

Partial sums of A320225.

Crossrefs

Cf. A001519, A022825, A025523, A033485, A078346, A320225, A351621.

Programs

  • Mathematica
    a[1] = 1; a[n_] := a[n] = a[n - 1] + Sum[a[Floor[n/k]], {k, 2, n}]; Table[a[n], {n, 1, 50}]

Formula

G.f. A(x) satisfies: A(x) = x/(1 - x) + (1/(1 - x)^2) * Sum_{k>=2} (1 - x^k) * A(x^k).

A351621 a(1) = 1; a(n) = 1 + a(n-1) + Sum_{k=2..n} a(floor(n/k)).

Original entry on oeis.org

1, 3, 6, 12, 19, 32, 46, 69, 96, 133, 171, 234, 298, 379, 471, 595, 720, 891, 1063, 1288, 1531, 1815, 2100, 2496, 2900, 3371, 3873, 4479, 5086, 5848, 6611, 7530, 8491, 9580, 10691, 12088, 13486, 15059, 16700, 18642, 20585, 22885, 25186, 27818, 30580, 33630, 36681, 40363, 44060, 48208
Offset: 1

Views

Author

Ilya Gutkovskiy, Feb 20 2022

Keywords

Comments

Partial sums of A345139.

Crossrefs

Cf. A001906, A022825, A025523, A078346, A102378, A345139, A351620.

Programs

  • Mathematica
    a[1] = 1; a[n_] := a[n] = 1 + a[n - 1] + Sum[a[Floor[n/k]], {k, 2, n}]; Table[a[n], {n, 1, 50}]

Formula

G.f. A(x) satisfies: A(x) = ( x + Sum_{k>=2} (1 - x^k) * A(x^k) ) / (1 - x)^2.
Showing 1-6 of 6 results.

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