A323753 Lesser member of primitive exponential amicable pairs.

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

Offset: 1

Author

Amiram Eldar, Jan 26 2019

Keywords

• nonn
• more

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.

Examples

			(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.
		

Links

• 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.

Crossrefs

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

Programs

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 *)