<p>We show the existence of a set <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(A\subseteq \mathbb {Z}_{\ge 2}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <mo>⊆</mo> <msub> <mi mathvariant="double-struck">Z</mi> <mrow> <mo>≥</mo> <mn>2</mn> </mrow> </msub> </mrow> </math></EquationSource> </InlineEquation> satisfying the estimates of the Bateman–Horn conjecture, Goldbach’s conjecture, and also <Equation ID="Equ12"> <EquationSource Format="TEX">\(\begin{aligned} \#\{p\le x \text { prime} ~|~ p\in A\} \gg x(\log \log x)/(\log x)^2. \end{aligned}\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <mo>#</mo> <mrow> <mo stretchy="false">{</mo> <mi>p</mi> <mo>≤</mo> <mi>x</mi> <mspace width="0.333333em" /> <mtext>prime</mtext> <mspace width="3.33333pt" /> <mo stretchy="false">|</mo> <mspace width="3.33333pt" /> <mi>p</mi> <mo>∈</mo> <mi>A</mi> <mo stretchy="false">}</mo> </mrow> <mo>≫</mo> <mi>x</mi> <mrow> <mo stretchy="false">(</mo> <mo>log</mo> <mo>log</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">/</mo> <msup> <mrow> <mo stretchy="false">(</mo> <mo>log</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </msup> <mo>.</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation></p>