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I tried the following:

sage: R = Zmod(7)
sage: for k in [2..500]:
....:     a = 2^k-1
....:     b = 2^(k-1)-1
....:     N = ZZ('{}{}'.format(a, b))
....:     if R(N) != R(6):
....:         continue
....:     print( "k=%s Is ec(k) prime? %s. Factorization follows:\nec(k) = %s\n"
....:            % (k, N.is_prime(), N.factor()) )
....:     
k=10 Is ec(k) prime? False. Factorization follows:
ec(k) = 19 * 103 * 523

k=11 Is ec(k) prime? False. Factorization follows:
ec(k) = 479 * 42737

k=14 Is ec(k) prime? False. Factorization follows:
ec(k) = 11 * 593 * 25117

k=28 Is ec(k) prime? False. Factorization follows:
ec(k) = 233 * 1607 * 716915680417

k=32 Is ec(k) prime? False. Factorization follows:
ec(k) = 131 * 4463 * 21601 * 44623 * 76213

k=49 Is ec(k) prime? False. Factorization follows:
ec(k) = 5 * 757 * 16333 * 1225015921 * 7433549000531

k=53 Is ec(k) prime? False. Factorization follows:
ec(k) = 5 * 337 * 53455188455436151040711945027

k=70 Is ec(k) prime? False. Factorization follows:
ec(k) = 109 * 839 * 75046613 * 241028036131 * 713694876516226387

k=71 Is ec(k) prime? False. Factorization follows:
ec(k) = 23 * 15737 * 65234886529801619745410789282584431073

k=74 Is ec(k) prime? False. Factorization follows:
ec(k) = 11 * 19 * 269 * 9532513 * 352463140718866450408093341421867

k=88 Is ec(k) prime? False. Factorization follows:
ec(k) = 31 * 73875972467027 * 135137137017690741456718218482342349371

k=92 Is ec(k) prime? False. Factorization follows:
ec(k) = 730315371175567 * 39625364799966331 * 1711101949753493724071011

k=109 Is ec(k) prime? False. Factorization follows:
ec(k) = 5 * 653 * 1606053961 * 10568312139584431 * 11711717200756188938696404879503826537

k=113 Is ec(k) prime? False. Factorization follows:
ec(k) = 5 * 193 * 149270993 * 24073195224569 * 29946980304751014175703201587995219695454299

k=130 Is ec(k) prime? False. Factorization follows:
ec(k) = 1129 * 959806273091 * 7986419296370382549203 * 157278660923445868781899626742322195007583

k=131 Is ec(k) prime? False. Factorization follows:
ec(k) = 51162479 * 2784303036149 * 567204394305177089 * 336916099985640327995882896303775632213517

k=148 Is ec(k) prime? False. Factorization follows:
ec(k) = 76292370683 * 46974096157407024149 * 60514961739327090714406687 * 1645269521635269788991753843968263

k=152 Is ec(k) prime? False. Factorization follows:
ec(k) = 31 * 107 * 677 * 2131 * 145361 * 173087 * 452931678706211 * 6576742625936687 * 159179030364736283121060312673245829294207

k=169 Is ec(k) prime? False. Factorization follows:
ec(k) = 5 * 67360183384144337 * 1317991479336685050851 * 1685712081595174413704015925364267420341119063462133679767171233

k=173 Is ec(k) prime? False. Factorization follows:
ec(k) = 5 * 1543 * 11287 * 1374911658072068607645715891596827336333789380835457971697269244791317892283158474011048315274579

k=190 Is ec(k) prime? False. Factorization follows:
ec(k) = 19 * 82593443886666852155734071357995610738188887427158348853883401985101228162919972298836892542211199706871473911269

and i had to stop here. (Since i did not see any sense in finding prime numbers of this "concatenated shape". This is non-structural mathematics for me.)