A126165 Smaller member of each exponential amicable pair.

90972, 454860, 937692, 1000692, 1182636, 1546524, 2092356, 2638188, 2820132, 3365964, 3729852, 3911796, 4275684, 4548600, 4688460, 4821516, 5003460, 5367348, 5549292, 5913180, 6095124, 6459012, 6640956, 7186788, 7550676, 7732620, 8096508, 8824284, 9188172, 9370116

Offset

1,1

Comments

This sequence includes the smallest member of all exponential amicable pairs and does not discriminate between primitive and nonprimitive pairs.

The asymptotic density of this sequence is Sum_{n>=1} f(A323753(n)) = 3.1138...*10^(-6), where f(n) = (6/(Pi^2*n)) * Product_{prime p|n} (p/(p+1)). - Amiram Eldar, Dec 05 2025

Links

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

Peter Hagis, Jr., Some results concerning exponential divisors, International Journal of Mathematics and Mathematical Sciences, Vol. 11, No. 2, (1988), pp. 343-349.

Ant King, Mathematica programs for A126164 - A126166

David Moews, Perfect, amicable and sociable numbers.

J. M. Pedersen, Known exponential amicable pairs.

Formula

The values of m for which esigma(m) = esigma(n) = m + n and m < n, where esigma is defined in A051377.

Example

a(3) = 937692 because (937692,968436) is the third exponential amicable pair.

Mathematica

fun[p_, e_] := DivisorSum[e, p^# &]; esigma[1] = 1; esigma[n_] := Times @@ fun @@@ FactorInteger[n]; s = {}; Do[m = esigma[n] - n; If[m > n && esigma[m] - m == n, AppendTo[s, n]], {n, 1, 10^7}]; s (* Amiram Eldar, May 09 2019 *)

Crossrefs

Cf. A051377, A054979, A049419, A054980, A126164, A323753.

Sequence in context: A212853 A127660 A127659 * A323753 A255985 A253838

Adjacent sequences: A126162 A126163 A126164 * A126166 A126167 A126168

Keyword

nonn

Author

Ant King, Dec 21 2006

Extensions

More terms from Amiram Eldar, May 09 2019

Status

approved