A160135 -id:A160135 - cp's OEIS frontend

A348627 Numbers that are both exponential and nonexponential abundant numbers.

3600, 4500, 6300, 7056, 8100, 8820, 9900, 14700, 21780, 22500, 25200, 30420, 31500, 35280, 39600, 46800, 49500, 52020, 56700, 58500, 61200, 61740, 64980, 68400, 69300, 76500, 77616, 81900, 82800, 85500, 88200, 89100, 91728, 95220, 97020, 103500, 104400, 105300

Offset: 1

Amiram Eldar, Oct 26 2021

nonn

			3600 is a term since A051377(3600) = 7920 > 2*3600 and A160135(3600) = 4573 > 3600.

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

Cf. A051377, A160135.
Intersection of A129575 and A348604.
Subsequence of A068403.
Similar sequence: A348523.

esigma[n_] := Times @@ (Sum[First[#]^d, {d, Divisors[Last[#]]}] &) /@ FactorInteger[n]; Select[Range[10^5], (e = esigma[#]) > 2*# && DivisorSigma[1, #] - e > # &]

A349178 Nonexponential harmonic numbers: numbers k that are not prime powers such that the harmonic mean of the nonexponential divisors of k is an integer.

Original entry on oeis.org

1645, 5742, 6336, 8925, 9450, 88473

Offset: 1

Amiram Eldar, Nov 09 2021

The prime powers are excluded since the primes and the squares of primes have a single nonexponential divisor (the number 1).

a(7) > 6.6*10^10, if it exists.

			1645 is a term since the set of its nonexponential divisors is {1, 5, 7, 35, 47, 235, 329} and the harmonic mean of this set, 5, is an integer.

Cf. A160097, A160135.

Similar sequences: A001599, A006086, A063947, A286325, A319745, A348964, A349026.

dQ[n_, m_] := (n > 0 && m > 0 && Divisible[n, m]); expDivQ[n_, d_] := Module[{ft = FactorInteger[n]}, And @@ MapThread[dQ, {ft[[;; , 2]], IntegerExponent[d, ft[[;; , 1]]]}]]; neDivs[1] = {0}; neDivs[n_] := Module[{d = Divisors[n]}, Select[d, ! expDivQ[n, #] &]]; Select[Range[10^4], Length[(d = neDivs[#])] > 1 && IntegerQ @ HarmonicMean[d] &]

cp's OEIS Frontend

A348627 Numbers that are both exponential and nonexponential abundant numbers.

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