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edited Oct 14, 2020 at 21:23

Brachylog, 22²² 16 bytes

-6 thanks to Zgarb

{|+₁|^₃;?-₁}ᶠḋᵐcF&<ḋF&<.ḋṀ;Fx×

ḋᵐcFḋF: all the prime factors of the three numbers are in list F.B^3-B
&<.: the output N is greater than B
ḋṀ: N's prime factors are at least 2
;Fx×: and the product of the prime factors without the ones in F unifies with the output N.

{|+₁|-₁}ᶠḋᵐcF&<.ḋṀ;Fx×

ḋᵐcF: all the prime factors of the three numbers are in list F.
&<.: the output N is greater than B
ḋṀ: N's prime factors are at least 2
;Fx×: and the product of the prime factors without the ones in F unifies with the output N.

-6 thanks to Zgarb

^₃;?-ḋF&<.ḋṀ;Fx×

^₃;?-ḋF: all the prime factors of B^3-B
&<.: the output N is greater than B
ḋṀ: N's prime factors are at least 2
;Fx×: and the product of the prime factors without the ones in F unifies with the output N.

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created Oct 14, 2020 at 15:27

{|+₁|-₁}ᶠḋᵐcF&<.ḋṀ;Fx×

{|+₁|-₁}ᶠ: [B,B+1,B-1]
ḋᵐcF: all the prime factors of the three numbers are in list F.
&<.: the output N is greater than B
ḋṀ: N's prime factors are at least 2
;Fx×: and the product of the prime factors without the ones in F unifies with the output N.