A383695 -id:A383695 - cp's OEIS frontend

A383696 Primitive exponential infinitary abundant numbers that are not primitive exponential unitary abundant: the powerful terms of A383695.

476985600, 815673600, 1018886400, 1177862400, 1493049600, 2014214400, 2373638400, 2712326400, 3756614400, 3863865600, 4744454400, 5218617600, 6234681600, 7928121600, 9824774400, 10502150400, 12669753600, 14227718400, 15040569600, 17614598400, 19443513600, 22356230400

Offset: 1

Amiram Eldar, May 06 2025

nonn

For squarefree numbers k, eusigma(k) = eisigma(k) = k, where eusigma is the sum of exponential unitary divisors function (A322857), and eisigma is the sum of exponential infinitary divisors function (A361175). Thus, if m is a term (eisigma(m) > 2*m >= eusigma(m)) and k is a squarefree number coprime to m, then eusigma(k*m) = eusigma(k) * eusigma(m) = k * eusigma(m) <= 2*k*m, and eisigma(k*m) = eisigma(k) * eisigma(m) = k * eisigma(m) > 2*k*m, so k*m is an exponential infinitary abundant number that is not exponential unitary abundant (A383695). Therefore, the sequence A383695 can be generated from this sequence by multiplying with coprime squarefree numbers.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Intersection of A001694 and A383695.

Cf. A005117, A322857, A361175.

idivs[1] = {1}; idivs[n_] := Sort @ Flatten @ Outer[Times, Sequence @@ (FactorInteger[n] /. {p_, e_Integer} :> p^Select[Range[0, e], BitOr[e, #] == e &])];
fi[p_, e_] := Total[p^idivs[e]]; fu[p_, e_] := DivisorSum[e, p^# &, CoprimeQ[#, e/#] &];
q[n_] := Module[{fct = FactorInteger[n]}, Times @@ fu @@@ fct <= 2*n < Times @@ fi @@@ fct];
pows[max_] := Union[Flatten[Table[i^2*j^3, {j, 1, Surd[max, 3]}, {i, 1, Sqrt[max/j^3]}]]];
seqA383696[max_] := Select[pows[max], q]; seqA383696[10^10]

isidiv(d, f) = {if (d==1, return (1)); for (k=1, #f~, bne = binary(f[k, 2]); bde = binary(valuation(d, f[k, 1])); if (#bde < #bne, bde = concat(vector(#bne-#bde), bde)); for (j=1, #bne, if (! bne[j] && bde[j], return (0)); ); ); return (1); } \\ Michel Marcus at A077609
fi(p, e) = sumdiv(e, d, if(isidiv(d, factor(e)), p^d, 0));
fu(p, e) = sumdiv(e, d, if(gcd(d, e/d)==1, p^d));
isprim(k) = {my(f = factor(k)); prod(i = 1, #f~, fu(f[i, 1], f[i, 2])) <= 2*k && prod(i = 1, #f~, fi(f[i, 1], f[i, 2])) > 2*k;}
listpows(lim) = my(v = List(), t); for(m = 1, sqrtnint(lim\1, 3), t=m^3; for(n = 1, sqrtint(lim\t), listput(v, t*n^2))); Set(v) \\ Charles R Greathouse IV at A001694
listA383696(lim) = select(x -> isprim(x), listpows(lim));

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

Amiram Eldar, Jan 11 2019

nonn

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

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eckford Cohen, A property of Dedekind's psi-function, Proceedings of the American Mathematical Society, Vol. 12, No. 6 (1961), p. 996.

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.

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 *)

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

cp's OEIS Frontend

A383696 Primitive exponential infinitary abundant numbers that are not primitive exponential unitary abundant: the powerful terms of A383695.

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