A323753 Lesser member of primitive exponential amicable pairs.

90972, 937692, 4548600, 44030448, 46884600, 453842928, 712931184, 906494400, 20907057600, 34793179200, 47646797328, 53469838800, 240707724300

Offset

1,1

Comments

Exponential amicable pair (m,n) is primitive if there is no prime number that is a unitary divisor of both m and n. All the other amicable pairs can be generated from primitive pairs by multiplying them with a squarefree integer coprime to each of the members of the pair. Hagis found the first 6 terms in 1988. Pedersen found the next 7 terms in 1999.

a(14) <= 588330137304.

The larger counterparts are in A323754.

Links

Table of n, a(n) for n=1..13.

Peter Hagis, Jr., Some Results Concerning Exponential Divisors, International Journal of Mathematics and Mathematical Sciences, Vol. 11, No. 2, (1988), pp. 343-350.

Jan Munch Pedersen, Known Exponential Amicable Pairs.

Example

(90972 = 2^2*3^2*7*19^2, 100548 = 2^2*3^3*7^2*19) are a primitive pair since they are an exponential amicable pair (A126165, A126166) and they do not have a common prime divisor with multiplicity 1 in both.
(454860, 502740) = 5 * (90972, 100548) are not a primitive pair since 5 divides both of them only once.

Mathematica

rad[n_] := Times @@ First /@ FactorInteger[n]; pf[n_] := Denominator[n/rad[n]^2]; esigma[n_] := Times @@ (Sum[First[#]^d, {d, Divisors[Last[#]]}] &) /@ FactorInteger[n]; es[n_] := esigma[n] - n; s = {}; Do[m = es[n]; If[m > n && es[m] == n && CoprimeQ[pf[n], pf[m]], AppendTo[s, n]], {n, 1, 10^7}]; s (* after Jean-François Alcover at A055231 and A051377 *)

Crossrefs

Cf. A051377, A054979, A049419, A054980, A126164, A126165, A126166, A126167, A323754.

Sequence in context: A127660 A127659 A126165 * A255985 A253838 A253845

Adjacent sequences: A323750 A323751 A323752 * A323754 A323755 A323756

Keyword

nonn,more

Author

Amiram Eldar, Jan 26 2019

Status

approved