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.

A383698 Primitive exponential squarefree exponential abundant numbers: the powerful terms of A383697.

Original entry on oeis.org

900, 1764, 4356, 4500, 4900, 6084, 10404, 12348, 12996, 19044, 30276, 34596, 44100, 47916, 49284, 60516, 66564, 79092, 79524, 88200, 101124, 108900, 112500, 125316, 132300, 133956, 152100, 161604, 176868, 181476, 191844, 213444, 217800, 220500, 224676, 246924
Offset: 1

Views

Author

Amiram Eldar, May 06 2025

Keywords

Comments

Subsequence of A383694 and first differs from it at n = 11.
The least odd term is a(1345) = 225450225, and the least term that is coprime to 6 is 1117347505588495206025.
For squarefree numbers k, essigma(k) = k, where essigma is the sum of exponential squarefree exponential divisors function (A361174). Thus, if m is a term (essigma(m) > 2*m) and k is a squarefree number coprime to m, then essigma(k*m) = essigma(k) * essigma(m) = k * essigma(m) > 2*k*m, so k*m is an exponential squarefree exponential abundant number. Therefore, the sequence of exponential squarefree exponential abundant numbers (A383697) can be generated from this sequence by multiplying with coprime squarefree numbers.

Links

Crossrefs

Intersection of A001694 and A383697.
Subsequence of A383694.
Cf. A005117, A361174.

Programs

  • Mathematica
    fun[p_, e_] := DivisorSum[e, p^# &, SquareFreeQ[#] &]; q[n_] := Min[(f = FactorInteger[n])[[;; , 2]]] > 1 && Times @@ fun @@@ f > 2*n; Select[Range[250000], q]
  • PARI
    fun(p, e) = sumdiv(e, d, if(issquarefree(d), p^d, 0));
isok(k) = {my(f = factor(k)); ispowerful(f) && prod(i = 1, #f~, fun(f[i, 1], f[i, 2])) > 2*k;}

A323332 The Dedekind psi function values of the powerful numbers, A001615(A001694(n)).

Original entry on oeis.org

1, 6, 12, 12, 24, 30, 36, 48, 72, 56, 96, 144, 108, 180, 216, 132, 150, 192, 288, 182, 336, 360, 432, 360, 324, 384, 576, 306, 648, 392, 380, 672, 720, 864, 672, 792, 900, 768, 552, 1152, 750, 1296, 1080, 1092, 972, 1344, 1440, 870, 1728, 2160, 992, 1584
Offset: 1

Views

Author

Amiram Eldar, Jan 11 2019

Keywords

Comments

The sum of the reciprocals of all the terms of this sequence is Pi^2/6 (A013661).
The asymptotic density of a sequence S that possesses the property that an integer k is a term if and only if its powerful part, A057521(k) is a term, is (1/zeta(2)) * Sum_{n>=1, A001694(n) is a term of S} 1/a(n). Examples for such sequences are the e-perfect numbers (A054979), the exponential abundant numbers (A129575), and other sequences listed in the Crossrefs section. - Amiram Eldar, May 06 2025

Links

Crossrefs

Cf. A001615, A001694, A013661, A082695, A112526.
Sequences whose density can be calculated using this sequence: A054979, A129575, A307958, A308053, A321147, A322858, A323310, A328135, A339936, A340109, A364990, A382061, A383693, A383695, A383697.

Programs

  • Mathematica
    psi[1]=1; psi[n_] := n * Times@@(1+1/Transpose[FactorInteger[n]][[1]]); psi /@ Join[{1}, Select[Range@ 1200, Min@ FactorInteger[#][[All, 2]] > 1 &]] (* after T. D. Noe at A001615 and Harvey P. Dale at A001694 *)
  • Python
    from math import isqrt, prod
from sympy import mobius, integer_nthroot, primefactors
def A323332(n):
    def squarefreepi(n): return int(sum(mobius(k)*(n//k**2) for k in range(1, isqrt(n)+1)))
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x):
        c, l = n+x-squarefreepi(integer_nthroot(x,3)[0]), 0
        j = isqrt(x)
        while j>1:
            k2 = integer_nthroot(x//j**2,3)[0]+1
            w = squarefreepi(k2-1)
            c -= j*(w-l)
            l, j = w, isqrt(x//k2**3)
        return c+l
    a = primefactors(m:=bisection(f,n,n))
    return m*prod(p+1 for p in a)//prod(a) # Chai Wah Wu, Sep 14 2024

A383693 Exponential unitary abundant numbers: numbers k such that A322857(k) > 2*k.

Original entry on oeis.org

900, 1764, 4356, 4500, 4900, 6084, 6300, 8820, 9900, 10404, 11700, 12348, 12996, 14700, 15300, 17100, 19044, 19404, 20700, 21780, 22500, 22932, 26100, 27900, 29988, 30276, 30420, 30492, 31500, 33300, 33516, 34596, 36900, 38700, 40572, 42300, 42588, 44100, 47700, 47916, 49284, 49500
Offset: 1

Views

Author

Amiram Eldar, May 05 2025

Keywords

Comments

First differs from its subsequence A383697 at n = 21.
All the terms are nonsquarefree numbers (A013929), since A322857(k) = k if k is a squarefree number (A005117).
If an exponential abundant number (A129575) is exponentially squarefree (A209061), then it is in this sequence. Terms of this sequence that are not exponentially squarefree are a(21) = 22500, a(77) = 86436, a(140) = 157500, etc..
The least odd term is a(202273) = 225450225, and the least term that is coprime to 6 is a(1.002..*10^18) = 1117347505588495206025.
The asymptotic density of this sequence is Sum_{n>=1} f(A383694(n)) = 0.00089722..., where f(n) = (6/(Pi^2*n))*Product_{prime p|n}(p/(p+1)).

Examples

			900 is a term since A322857(900) = 2160 > 2*900 = 1800.

Links

Crossrefs

Cf. A005117, A209061, A322857, A322858, A361255.
Subsequence of A013929 and A129575.
Subsequences: A383694, A383697, A383698.

Programs

  • Mathematica
    f[p_, e_] := DivisorSum[e, p^# &, GCD[#, e/#] == 1 &]; q[n_] := Times @@ f @@@ FactorInteger[n] > 2 n; Select[Range[50000], q]
  • PARI
    fun(p, e) = sumdiv(e, d, if(gcd(d, e/d) == 1, p^d));
isok(k) = {my(f = factor(k)); prod(i = 1, #f~, fun(f[i, 1], f[i, 2])) > 2*k;}
Showing 1-3 of 3 results.

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