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.

A183097 a(n) = sum of powerful divisors d (including 1) of n.

Original entry on oeis.org

1, 1, 1, 5, 1, 1, 1, 13, 10, 1, 1, 5, 1, 1, 1, 29, 1, 10, 1, 5, 1, 1, 1, 13, 26, 1, 37, 5, 1, 1, 1, 61, 1, 1, 1, 50, 1, 1, 1, 13, 1, 1, 1, 5, 10, 1, 1, 29, 50, 26, 1, 5, 1, 37, 1, 13, 1, 1, 1, 5, 1, 1, 10, 125, 1, 1, 1, 5, 1, 1, 1, 130, 1, 1, 26, 5, 1, 1, 1, 29, 118, 1, 1, 5, 1, 1, 1, 13, 1, 10, 1, 5, 1, 1, 1, 61, 1, 50, 10, 130
Offset: 1

Views

Author

Jaroslav Krizek, Dec 25 2010

Keywords

Comments

Sequence is not the same as A091051(n); a(72) = 130, A091051(72) = 58.
a(n) = sum of divisors d of n from set A001694 - powerful numbers.

Examples

			For n = 12, set of such divisors is {1, 4}; a(12) = 1+4 = 5.
		

Crossrefs

Programs

  • Maple
    A183097 := proc(n)
        local a,pe,p,e ;
        a := 1;
        for pe in ifactors(n)[2] do
            p := op(1,pe) ;
            e := op(2,pe) ;
            if e > 1 then
                a := a* ( (p^(e+1)-1)/(p-1)-p) ;
            end if;
        end do:
        a ;
    end proc: # R. J. Mathar, Jun 02 2020
  • Mathematica
    fun[p_,e_] := (p^(e+1)-1)/(p-1) - p; a[1] = 1; a[n_] := Times @@ (fun @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, May 14 2019 *)
  • PARI
    A183097(n) = sumdiv(n, d, ispowerful(d)*d); \\ Antti Karttunen, Oct 07 2017
    
  • PARI
    a(n) = {my(f = factor(n)); prod(i = 1, #f~, (f[i,1]^(f[i,2]+1)-1) / (f[i,1]-1) - f[i,1]);} \\ Amiram Eldar, Dec 24 2023

Formula

a(n) = A000203(n) - A183098(n) = A183100(n) + 1.
a(1) = 1, a(p) = 1, a(pq) = 1, a(pq...z) = 1, a(p^k) = ((p^(k+1)-1) / (p-1))-p, for p, q = primes, k = natural numbers, pq...z = product of k (k > 2) distinct primes p, q, ..., z.
From Amiram Eldar, Dec 24 2023: (Start)
Dirichlet g.f.: zeta(s) * zeta(2*s-2) * zeta(3*s-3) / zeta(6*s-6).
Sum_{k=1..n} a(k) ~ c * n^(3/2), where c = zeta(3/2)^2/(3*zeta(3)) = 1.892451... . (End)

A183098 a(1) = 0, a(n) = sum of divisors d of n such that if d = Product_{i} (p_i^e_i) then not all e_i are > 1.

Original entry on oeis.org

0, 2, 3, 2, 5, 11, 7, 2, 3, 17, 11, 23, 13, 23, 23, 2, 17, 29, 19, 37, 31, 35, 23, 47, 5, 41, 3, 51, 29, 71, 31, 2, 47, 53, 47, 41, 37, 59, 55, 77, 41, 95, 43, 79, 68, 71, 47, 95, 7, 67, 71, 93, 53, 83, 71, 107, 79, 89, 59, 163, 61, 95, 94, 2, 83, 143, 67, 121, 95, 143, 71, 65, 73, 113, 98, 135, 95, 167, 79, 157, 3, 125, 83, 219, 107, 131, 119, 167, 89, 224, 111, 163, 127, 143, 119, 191, 97, 121, 146, 87
Offset: 1

Views

Author

Jaroslav Krizek, Dec 25 2010

Keywords

Comments

a(n) = sum of non-powerful divisors d of n where powerful numbers are numbers from A001694(m) for m >= 2.
Sequence is not the same as A183101(n): a(72) = 65, A183101(72) = 137.

Examples

			For n = 12, the set of such divisors is {2, 3, 6, 12}; a(12) = 2+3+6+12 = 23.
		

Crossrefs

Programs

  • Mathematica
    f1[p_, e_] := (p^(e+1)-1)/(p-1); f2[p_, e_] := f1[p, e] - p; a[1] = 0; a[n_] := Times @@ f1 @@@ (f = FactorInteger[n]) - Times @@ f2 @@@ f; Array[a, 100] (* Amiram Eldar, Aug 29 2023 *)
  • PARI
    A183098(n) = sumdiv(n, d, d*(!ispowerful(d))); \\ Antti Karttunen, Oct 07 2017

Formula

a(n) = A000203(n) - A183097(n) = A183100(n) - 1.
a(1) = 0, a(p) = p, a(p*q) = p+q+p*q, a(p*q*...*z) = (p+1)*(q+1)*...*(z+1) - 1, a(p^k) = p, for p, q = primes, k = natural numbers, p*q*...*z = product of k (k > 2) distinct primes p, q, ..., z.

Extensions

Name corrected by Jon E. Schoenfield, Aug 29 2023

A183099 a(n) = sum of powerful divisors d (excluding 1) of n.

Original entry on oeis.org

0, 0, 0, 4, 0, 0, 0, 12, 9, 0, 0, 4, 0, 0, 0, 28, 0, 9, 0, 4, 0, 0, 0, 12, 25, 0, 36, 4, 0, 0, 0, 60, 0, 0, 0, 49, 0, 0, 0, 12, 0, 0, 0, 4, 9, 0, 0, 28, 49, 25, 0, 4, 0, 36, 0, 12, 0, 0, 0, 4, 0, 0, 9, 124, 0, 0, 0, 4, 0, 0, 0, 129, 0, 0, 25, 4, 0, 0, 0, 28, 117, 0, 0, 4, 0, 0, 0, 12, 0, 9, 0, 4, 0, 0, 0, 60, 0, 49, 9, 129
Offset: 1

Views

Author

Jaroslav Krizek, Dec 25 2010

Keywords

Comments

a(n) = sum of divisors d of n from set A001694(m) - powerful numbers for m >=2.

Examples

			For n = 12, set of such divisors is {4}; a(12) = 4.
		

Crossrefs

Programs

  • Mathematica
    f[p_, e_] := (p^(e+1)-1)/(p-1) - p; a[1] = 0; a[n_] := Times @@ f @@@ FactorInteger[n] - 1; Array[a, 100] (* Amiram Eldar, Aug 29 2023 *)
  • PARI
    A183099(n) = (sumdiv(n, d, ispowerful(d)*d) - 1); \\ Antti Karttunen, Oct 07 2017

Formula

a(n) = A000203(n) - A183100(n) = A183097(n) - 1.
a(1) = 0, a(p) = 0, a(pq) = 0, a(pq...z) = 0, a(p^k) = ((p^(k+1)-1) / (p-1))-p-1, for p, q = primes, k = natural numbers, pq...z = product of k (k > 2) distinct primes p, q, ..., z.
Showing 1-3 of 3 results.