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.

A323310 List of e-unitary perfect numbers that are not e-semiproper perfect numbers.

Original entry on oeis.org

4769856, 23849280, 52468416, 81087552, 90627264, 109706688, 138325824, 147865536, 176484672, 195564096, 205103808, 224183232, 252802368, 262342080, 281421504, 290961216, 319580352, 338659776, 348199488, 357739200, 376818624, 395898048, 405437760, 424517184
Offset: 1

Views

Author

Amiram Eldar, Jan 10 2019

Keywords

Comments

The e-unitary perfect numbers are numbers k such that the sum of their exponential unitary divisors (A322857) equals 2k. The e-semiproper perfect numbers are numbers k such that the sum of their exponential semiproper divisors (A323309) equals 2k. Apparently most of the e-unitary perfect numbers are also e-semiproper perfect numbers: The first 41393 e-unitary perfect numbers are also the first 41393 e-semiproper perfect numbers, but the 41394th e-unitary perfect number is 4769856 which is not e-semiproper perfect. This number, which is the first term of this sequence, was found by Minculete.
The powerful (A001694) terms of this sequence are the primitive terms, i.e., if k is a powerful term, then m*k is a term for any squarefree (A005117) number m that is coprime to k. The only primitive terms below 10^18 are 4769856 and 357739200. If S is the sequence of primitive terms, then the asymptotic density of this sequence is Sum_{n>=1} f(S(n)) = 5.235...*10^(-8), where f(n) = (6/(Pi^2*n))*Product_{prime p|n}(p/(p+1)). - Amiram Eldar, May 06 2025

Links

Crossrefs

Cf. A322857, A322858, A323308, A323309.

Programs

  • Mathematica
    fs[p_, e_] := If[e==1, p, p^e + p]; a[1]=1; essigma[n_] := Times @@ fs @@@ FactorInteger[n]; esPerfectQ[n_] := essigma[n]==2n; fu[p_, e_] := DivisorSum[e, p^# &, GCD[#, e/#]==1 &]; eusigma[n_] := Times @@ fu @@@ FactorInteger[n]; euPerfectQ[n_] := eusigma[n] == 2n; aQ[n_] := euPerfectQ[n] && !esPerfectQ[n]; Select[Range[1, 10^8], aQ]

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;}

A349026 Exponential unitary harmonic numbers: numbers k such that the harmonic mean of the exponential unitary divisors of k is an integer.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 12, 13, 14, 15, 17, 18, 19, 21, 22, 23, 26, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 46, 47, 51, 53, 55, 57, 58, 59, 60, 61, 62, 65, 66, 67, 69, 70, 71, 73, 74, 75, 77, 78, 79, 82, 83, 84, 85, 86, 87, 89, 90, 91, 93, 94
Offset: 1

Views

Author

Amiram Eldar, Nov 06 2021

Keywords

Comments

First differs from A348964 at n = 102. a(102) = 144 is not an exponential harmonic number of type 2.
The exponential unitary divisors of n = Product p(i)^e(i) are all the numbers of the form Product p(i)^b(i) where b(i) is a unitary divisor of e(i) (see A278908).
Equivalently, numbers k such that A349025(k) | k * A278908(k).

Examples

			The squarefree numbers are trivial terms. If k is squarefree, then it has a single exponential unitary divisor, k itself, and thus the harmonic mean of its exponential unitary divisors is also k, which is an integer.
144 is a term since its exponential unitary divisors are 6, 18, 48 and 144, and their harmonic mean, 16, is an integer.

Links

Crossrefs

Cf. A278908 (number of exponential unitary divisors), A322857, A322858, A323310, A349025, A349027.
Similar sequences: A001599, A006086, A063947, A286325, A319745, A348964.

Programs

  • Mathematica
    f[p_, e_] := p^e * 2^PrimeNu[e] / DivisorSum[e, p^(e - #) &, CoprimeQ[#, e/#] &]; euhQ[1] = True; euhQ[n_] := IntegerQ[Times @@ f @@@ FactorInteger[n]]; Select[Range[100], euhQ]
Showing 1-4 of 4 results.

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