(MAGMAMagma) sol:=[]; for n in [1..70] do k:=n; while not IsPrime(3^n+3^k-1) and k gt 0 do k:=k-1; end while; if k ge 0 then Append(~sol, k); else Append(~sol, 0); end if; end for; sol; // Marius A. Burtea, Sep 16 2019
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For n = 154, for k = 0, 1, 2, 3, ... , , 154, the numbers 3^154 + 3^k - 1 are respectively divisible by 3, 5531, 73, 5, 7, 19001, 6553, 5, 239, 3541, 7, 5, 33247, 71, 19, 5, 7, 29, 1973, 5, 436467739, 71161, 7, 5, 4283, 37, 73, 5, 7, 11177, 13721, 5, 19, 29207, 7, 5, 64849, 4001, 73, 5, 7, 31, 227009113, 5, 139, 29, 7, 5, 71, 107102231, 19, 5, 7, 10765021647412056860623883, 521, 5, 241, 1448445976112887644909473, 7, 5, 5657, 37, 73, 5, 7, 110661029, 65963, 5, 19, 20411, 7, 5, 331, 29, 73, 5, 7, 7671791, 269, 5, 563, 211, 7, 5, 6553, 113, 19, 5, 7, 425679689, 1301, 5, 334244063, 53, 7, 5, 15254167, 37, 73, 5, 7, 29, 3391, 5, 19, 7727, 7, 5, 4799, 2269, 73, 5, 7, 1822597729, 47, 5, 242496218184092003, 38971, 7, 5, 9994245487379630507640393493999, 9104413, 19, 5, 7, 3581, 1039, 5, 181, 29, 7, 5, 2472681219552827727900539, 37, 73, 5, 7, 47, 99986141, 5, 19, 1237, 7, 5, 55817, 53, 73, 5, 7, 1033, 9187, 5, 199, 71, 7. So a(154) = 0.
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If For n = 154, for k = 0, 1, 2, 3, ... , 154, the numbers 3^154 + 3^k - 1 are respectively divisible by 3, 5531, 73, 5, 7, 19001, 6553, 5, 239, 3541, 7, 5, 33247, 71, 19, 5, 7, 29, 1973, 5, 436467739, 71161, 7, 5, 4283, 37, 73, 5, 7, 11177, 13721, 5, 19, 29207, 7, 5, 64849, 4001, 73, 5, 7, 31, 227009113, 5, 139, 29, 7, 5, 71, 107102231, 19, 5, 7, 10765021647412056860623883, 521, 5, 241, 1448445976112887644909473, 7, 5, 5657, 37, 73, 5, 7, 110661029, 65963, 5, 19, 20411, 7, 5, 331, 29, 73, 5, 7, 7671791, 269, 5, 563, 211, 7, 5, 6553, 113, 19, 5, 7, 425679689, 1301, 5, 334244063, 53, 7, 5, 15254167, 37, 73, 5, 7, 29, 3391, 5, 19, 7727, 7, 5, 4799, 2269, 73, 5, 7, 1822597729, 47, 5, 242496218184092003, 38971, 7, 5, 9994245487379630507640393493999, 9104413, 19, 5, 7, 3581, 1039, 5, 181, 29, 7, 5, 2472681219552827727900539, 37, 73, 5, 7, 47, 99986141, 5, 19, 1237, 7, 5, 55817, 53, 73, 5, 7, 1033, 9187, 5, 199, 71, 7. So a(154) = 0.
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If n = 154, for k = 0, 1, 2, 3, ... , 154, the numbers 3^154 + 3^k - 1 are respectively divisible by 3, 5531, 73, 5, 7, 19001, 6553, 5, 239, 3541, 7, 5, 33247, 71, 19, 5, 7, 29, 1973, 5, 436467739, 71161, 7, 5, 4283, 37, 73, 5, 7, 11177, 13721, 5, 19, 29207, 7, 5, 64849, 4001, 73, 5, 7, 31, 227009113, 5, 139, 29, 7, 5, 71, 107102231, 19, 5, 7, 10765021647412056860623883, 521, 5, 241, 1448445976112887644909473, 7, 5, 5657, 37, 73, 5, 7, 110661029, 65963, 5, 19, 20411, 7, 5, 331, 29, 73, 5, 7, 7671791, 269, 5, 563, 211, 7, 5, 6553, 113, 19, 5, 7, 425679689, 1301, 5, 334244063, 53, 7, 5, 15254167, 37, 73, 5, 7, 29, 3391, 5, 19, 7727, 7, 5, 4799, 2269, 73, 5, 7, 1822597729, 47, 5, 242496218184092003, 38971, 7, 5, 9994245487379630507640393493999, 9104413, 19, 5, 7, 3581, 1039, 5, 181, 29, 7, 5, 2472681219552827727900539, 37, 73, 5, 7, 47, 99986141, 5, 19, 1237, 7, 5, 55817, 53, 73, 5, 7, 1033, 9187, 5, 199, 71, 7. So a(154) = 0.