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 1-3 of 3 results.

A025523 a(n) = 1 + Sum_{ k < n and k | n} a(k).

Original entry on oeis.org

1, 2, 3, 5, 6, 9, 10, 14, 16, 19, 20, 28, 29, 32, 35, 43, 44, 52, 53, 61, 64, 67, 68, 88, 90, 93, 97, 105, 106, 119, 120, 136, 139, 142, 145, 171, 172, 175, 178, 198, 199, 212, 213, 221, 229, 232, 233, 281, 283, 291, 294, 302, 303, 323, 326, 346, 349, 352, 353, 397, 398, 401
Offset: 1

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Author

David W. Wilson

Keywords

Comments

Permanent of n X n (0,1) matrix defined by A(i,j)=1 iff j=1 or i divides j. - Vladeta Jovovic, Jul 05 2003
Partial sums of A074206, ordered factorizations. - Augustine O. Munagi, Jul 10 2007
The subsequence of primes begins: 2, 3, 5, 19, 29, 43, 53, 61, 67, 97, 139, 199, 229, 233, 281, 283, 349, 353, 397, 401. - Jonathan Vos Post

Links

Crossrefs

Cf. A002321, A022825, A347031.
Cf. A074206, A002033.
A173382 is an essentially identical sequence.

Programs

  • Mathematica
    f[1] = 1; f[n_] := f[n] = DivisorSum[n, f[#] &, # < n &]; Accumulate[Array[f, 100]] (* Amiram Eldar, May 02 2025 *)
  • Python
    from functools import lru_cache
@lru_cache(maxsize=None)
def A025523(n):
    if n == 0:
        return 1
    c, j = 2, 2
    k1 = n//j
    while k1 > 1:
        j2 = n//k1 + 1
        c += (j2-j)*A025523(k1)
        j, k1 = j2, n//j2
    return n+c-j # Chai Wah Wu, Mar 30 2021

Formula

Running sum of A002033.
a(n) = 1 + Sum_{k = 2 to n} a([n/k]). - Mitchell Lee (worthawholephan(AT)gmail.com), Jul 26 2010
G.f. A(x) satisfies: A(x) = (1/(1 - x)) * (x + Sum_{k>=2} (1 - x^k) * A(x^k)). - Ilya Gutkovskiy, Aug 11 2021

A308077 G.f. A(x) satisfies: A(x) = x - A(x^2) + A(x^3) - A(x^4) + A(x^5) - A(x^6) + ...

Original entry on oeis.org

1, -1, 1, 0, 1, -3, 1, 0, 2, -3, 1, 2, 1, -3, 3, 0, 1, -8, 1, 2, 3, -3, 1, 0, 2, -3, 4, 2, 1, -13, 1, 0, 3, -3, 3, 10, 1, -3, 3, 0, 1, -13, 1, 2, 8, -3, 1, 0, 2, -8, 3, 2, 1, -20, 3, 0, 3, -3, 1, 18, 1, -3, 8, 0, 3, -13, 1, 2, 3, -13, 1, -4, 1, -3, 8, 2, 3, -13, 1
Offset: 1

Views

Author

Ilya Gutkovskiy, May 11 2019

Keywords

Links

Crossrefs

Cf. A006005 (positions of 1's), A067856, A307776, A347031.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n<2, n, add(a(n/d)*
     (-1)^(d-1), d=numtheory[divisors](n) minus {1}))
    end:
seq(a(n), n=1..80);  # Alois P. Heinz, Mar 30 2023
  • Mathematica
    terms = 79; A[] = 0; Do[A[x] = x + Sum[(-1)^(k + 1) A[x^k], {k, 2, terms}] + O[x]^(terms + 1) // Normal, terms + 1]; Rest[CoefficientList[A[x], x]]
a[n_] := If[n == 1, n, Sum[If[d < n, (-1)^(n/d + 1) a[d], 0], {d, Divisors[n]}]]; Table[a[n], {n, 1, 79}]

Formula

a(1) = 1; a(n) = Sum_{d|n, d

A347030 a(n) = 1 + Sum_{k=2..n} (-1)^k * a(floor(n/k)).

Original entry on oeis.org

1, 2, 1, 3, 2, 1, 0, 4, 4, 3, 2, 0, -1, -2, -1, 7, 6, 6, 5, 3, 4, 3, 2, -2, -2, -3, -3, -5, -6, -5, -6, 10, 11, 10, 11, 11, 10, 9, 10, 6, 5, 6, 5, 3, 3, 2, 1, -7, -7, -7, -6, -8, -9, -9, -8, -12, -11, -12, -13, -11, -12, -13, -13, 19, 20, 21, 20, 18, 19, 20, 19, 19, 18, 17, 17
Offset: 1

Views

Author

Ilya Gutkovskiy, Aug 11 2021

Keywords

Comments

Partial sums of A067856.

Links

Crossrefs

Cf. A002321, A025523, A067856, A309288, A347031.

Programs

  • Mathematica
    a[n_] := a[n] = 1 + Sum[(-1)^k a[Floor[n/k]], {k, 2, n}]; Table[a[n], {n, 1, 75}]
nmax = 75; A[] = 0; Do[A[x] = (1/(1 - x)) (x + Sum[(-1)^k (1 - x^k) A[x^k], {k, 2, nmax}]) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] // Rest
  • Python
    from functools import lru_cache
@lru_cache(maxsize=None)
def A347030(n):
    if n <= 1:
        return n
    c, j = 1, 2
    k1 = n//j
    while k1 > 1:
        j2 = n//k1 + 1
        c += (j2-j&1)*(-1 if j&1 else 1)*A347030(k1)
        j, k1 = j2, n//j2
    return c+(n+1-j&1)*(-1 if j&1 else 1) # Chai Wah Wu, Apr 04 2023

Formula

G.f. A(x) satisfies: A(x) = (1/(1 - x)) * (x + Sum_{k>=2} (-1)^k * (1 - x^k) * A(x^k)).
Showing 1-3 of 3 results.

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