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another problem with Infinite Products


Gary Adamson a long time ago came up with a sort of ultra-zeta sum:
A=Sum[1/n^n,{n,1,Infinity}]=1.291..
That is very close to  2- Sqrt[2]...
One wonders if the Product:
Product[1/(1-1/Prime[n]^Prime[n]),{n,1,Infinity}]-->A
or is it?
Product[1/(1-1/Prime[n]^n),{n,1,Infinity}]-->A
It is clear that both these products should converge
and I think that the second should be A on reconsideration.
But Mathematica disagrees:

A = Sum[1/n^n, {n, 1, Infinity}]
N[%]
1.2912859970626636`
Product[1/(1 - 1/Prime[n]^Prime[n]), {n, 1, Infinity}]
N[%]
1.3850602852044895`
Product[1/(1-1/Prime[n]^n),{n,1,Infinity}]
N[%]
2.2691047868959395`


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