Theorem: Numbers for which Euler Phi Function of 2n + 1 is less than that of 2n

The sequence of positive integers for which:
:$\map \phi {2 n + 1} < \map \phi {2 n}$
begins:
:$157, 262, 367, 412, \ldots$
{{OEIS|A001837}}

Proof:
{{begin-eqn}}
{{eqn | l = 2 \times 157
      | r = 314
      | c = 
}}
{{eqn | ll= \leadsto
      | l = \map \phi {2 \times 157}
      | r = 156
      | c = {{EulerPhiLink|314}}
}}
{{eqn | l = 2 \times 157 + 1
      | r = 315
      | c = 
}}
{{eqn | ll= \leadsto
      | l = \map \phi {2 \times 157 + 1}
      | r = 144
      | c = {{EulerPhiLink|315}}
}}
{{eqn | ll= \leadsto
      | l = \map \phi {2 \times 157 + 1}
      | o = <
      | r = \map \phi {2 \times 157}
      | c = 
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 2 \times 262
      | r = 524
      | c = 
}}
{{eqn | ll= \leadsto
      | l = \map \phi {2 \times 262}
      | r = 260
      | c = {{EulerPhiLink|524}}
}}
{{eqn | l = 2 \times 262 + 1
      | r = 525
      | c = 
}}
{{eqn | ll= \leadsto
      | l = \map \phi {2 \times 262 + 1}
      | r = 240
      | c = {{EulerPhiLink|525}}
}}
{{eqn | ll= \leadsto
      | l = \map \phi {2 \times 262 + 1}
      | o = <
      | r = \map \phi {2 \times 262}
      | c = 
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 2 \times 367
      | r = 734
      | c = 
}}
{{eqn | ll= \leadsto
      | l = \map \phi {2 \times 367}
      | r = 366
      | c = {{EulerPhiLink|734}}
}}
{{eqn | l = 2 \times 367 + 1
      | r = 735
      | c = 
}}
{{eqn | ll= \leadsto
      | l = \map \phi {2 \times 367 + 1}
      | r = 336
      | c = {{EulerPhiLink|735}}
}}
{{eqn | ll= \leadsto
      | l = \map \phi {2 \times 367 + 1}
      | o = <
      | r = \map \phi {2 \times 367}
      | c = 
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 2 \times 412
      | r = 824
      | c = 
}}
{{eqn | ll= \leadsto
      | l = \map \phi {2 \times 412}
      | r = 408
      | c = {{EulerPhiLink|824}}
}}
{{eqn | l = 2 \times 412 + 1
      | r = 825
      | c = 
}}
{{eqn | ll= \leadsto
      | l = \map \phi {2 \times 412 + 1}
      | r = 400
      | c = {{EulerPhiLink|825}}
}}
{{eqn | ll= \leadsto
      | l = \map \phi {2 \times 412 + 1}
      | o = <
      | r = \map \phi {2 \times 412}
      | c = 
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 2 \times 472
      | r = 944
      | c = 
}}
{{eqn | ll= \leadsto
      | l = \map \phi {2 \times 472}
      | r = 464
      | c = {{EulerPhiLink|944}}
}}
{{eqn | l = 2 \times 472 + 1
      | r = 945
      | c = 
}}
{{eqn | ll= \leadsto
      | l = \map \phi {2 \times 472 + 1}
      | r = 432
      | c = {{EulerPhiLink|945}}
}}
{{eqn | ll= \leadsto
      | l = \map \phi {2 \times 472  + 1}
      | o = <
      | r = \map \phi {2 \times 472}
      | c = 
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 2 \times 487
      | r = 974
      | c = 
}}
{{eqn | ll= \leadsto
      | l = \map \phi {2 \times 487}
      | r = 486
      | c = {{EulerPhiLink|974}}
}}
{{eqn | l = 2 \times 487 + 1
      | r = 975
      | c = 
}}
{{eqn | ll= \leadsto
      | l = \map \phi {2 \times 487 + 1}
      | r = 480
      | c = {{EulerPhiLink|975}}
}}
{{eqn | ll= \leadsto
      | l = \map \phi {2 \times 487 + 1}
      | o = <
      | r = \map \phi {2 \times 487}
      | c = 
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 2 \times 577
      | r = 1154
      | c = 
}}
{{eqn | ll= \leadsto
      | l = \map \phi {2 \times 577}
      | r = 576
      | c = {{EulerPhiLink|1154}}
}}
{{eqn | l = 2 \times 577 + 1
      | r = 1155
      | c = 
}}
{{eqn | ll= \leadsto
      | l = \map \phi {2 \times 577 + 1}
      | r = 480
      | c = {{EulerPhiLink|1155}}
}}
{{eqn | ll= \leadsto
      | l = \map \phi {2 \times 577 + 1}
      | o = <
      | r = \map \phi {2 \times 577}
      | c = 
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 2 \times 682
      | r = 1364
      | c = 
}}
{{eqn | ll= \leadsto
      | l = \map \phi {2 \times 682}
      | r = 600
      | c = {{EulerPhiLink|1364}}
}}
{{eqn | l = 2 \times 682 + 1
      | r = 1365
      | c = 
}}
{{eqn | ll= \leadsto
      | l = \map \phi {2 \times 682 + 1}
      | r = 576
      | c = {{EulerPhiLink|1365}}
}}
{{eqn | ll= \leadsto
      | l = \map \phi {2 \times 682 + 1}
      | o = <
      | r = \map \phi {2 \times 682}
      | c = 
}}
{{end-eqn}}
{{ProofWanted|that these are the smallest such. The philosophical significance of this result is not immediately clear.}}