If p is a sturdy prime, then p satisfies this property: * For all x>log_2(p), 1+A000120(p-(2^x mod p)) >= A000120(p). Almost all primes do not satisfy this property. Here is a list of all primes < 5*10^12 that satisfy the property, and whether each prime is sturdy. Prime p A000120(p) Sturdy? Comments 2 1 yes a(1) 3 2 yes a(2) 5 2 yes a(3) 7 3 yes a(4) 17 2 yes a(5) 31 5 yes a(6) 73 3 yes a(7) 89 4 yes a(8) 127 7 yes a(9) 257 2 yes a(10) 1801 5 yes a(11) 2089 4 yes a(12) 2593 4 no Divisor of 1 + 2^27 + 2^54 4177 4 no Divisor of 1 + 2^16 + 2^49 8191 13 yes a(13) 8713 4 no Divisor of 1 + 2^101 + 2^210 49201 5 no Divisor of 1 + 2^26 + 2^89 49297 5 no Divisor of 1 + 2^79 + 2^158 65537 2 yes a(14) 122921 7 no Divisor of 270549121 131071 17 yes a(15) 178481 9 yes a(16) 262151 4 no Divisor of 1 + 2^498 + 2^962 262337 4 no Divisor of 1 + 2^238 + 2^396 262433 4 no Divisor of 1 + 2^115 + 2^1105 262657 3 yes a(17) 266369 4 no Divisor of 1 + 2^60 + 2^495 458897 6 no Divisor of 1 + 2^16 + 2^64 + 2^126 524287 19 yes a(18) 540689 4 no Divisor of 1 + 2^492 + 2^1532 1121297 7 no Divisor of 1 + 2^10 + 2^24 + 2^37 + 2^75 2099863 8 yes a(19) 13264529 11 no Divisor of 331613225 13960201 8 no Divisor of 1 + 2^53 + 2^106 25165897 5 no Divisor of 1 + 2^631 + 2^3723 75497761 5 no Divisor of 1 + 2^8591 + 2^9048 212885833 11 no Divisor of 13744973807645 537460753 5 no Divisor of 1 + 2^26081 + 2^54299 538001921 6 no Divisor of 1 + 2^15241 + 2^27400 540701761 11 no Divisor of 1 + 2^53 + 2^106 545259553 4 no Divisor of 1 + 2^605 + 2^23178 554190913 7 no Divisor of 1 + 2^5583 + 2^11166 616318177 13 yes a(20) 672403969 7 no Divisor of 1 + 2^1011 + 2^2022 706741001 8 no Divisor of 1 + 2^6 + 2^1330 + 2^1398 1075839001 5 no Divisor of 1 + 2^10640 + 2^59141 1117808897 9 no Divisor of 1 + 2^187 + 2^237 + 2^346 + 2^383 2147483647 31 yes a(21) 2148532241 4 no Divisor of 1 + 2^57664 + 2^67006 2684494081 6 no Divisor of 1 + 2^20552 + 2^21846 4294967561 4 no Divisor of 1 + 2^68560 + 2^105188 4295229761 5 no Divisor of 1 + 2^2264 + 2^28375 5639328001 9 no Divisor of 1 + 2^247 + 2^1316 + 2^3147 6997164553 9 no Divisor of 1 + 2^316 + 2^1864 + 2^2167 8590065793 4 no Divisor of 1 + 2^17037 + 2^70355 8606728321 5 no Divisor of 1 + 2^72250 + 2^73310 17180131849 5 no Divisor of 1 + 2^198296 + 2^202061 17721197569 6 no Divisor of 1 + 2^22410 + 2^156064 34460406289 7 no Divisor of 1 + 2^116449 + 2^214546 34493964353 5 no Divisor of 1 + 2^65133 + 2^109300 60726444167 13 no Divisor of 2550935740123169 68719608833 4 no Divisor of 1 + 2^346406 + 2^430138 158913790081 5 no Divisor of 1 + 2^133586 + 2^427973 274894684417 4 no Divisor of 1 + 2^203712 + 2^931231 279726800897 7 no Divisor of 1 + 2^29801 + 2^117018 414808412209 11 no Divisor of 37779526340308081573953 549789385217 5 no Divisor of 1 + 2^165921 + 2^892440 577673101321 6 no Divisor of 1 + 2^104268 + 2^623040 761838257287 16 no Divisor of 355778466153029 1099553571073 5 no Divisor of 1 + 2^8608 + 2^66953 1101668812801 7 no Divisor of 1 + 2^525599 + 2^1571316 1168233267241 7 no Divisor of 1 + 2^469745 + 2^939490 2199027454081 5 no Divisor of 1 + 2^506590 + 2^1207721 4398050709761 5 no Divisor of 1 + 2^279063 + 2^1554493 4399258800137 8 no Divisor of 1 + 2^43 + 2^3970 + 2^4229 + 2^4790 4402493063201 8 no Divisor of 1 + 2^5299 + 2^10862 + 2^23117 4410931413073 6 no Divisor of 1 + 2^3716796 + 2^3886080 4432676798593 7 yes a(22) 4468913477761 7 no Divisor of 1 + 2^2318149 + 2^2522095 4537903505537 9 no Divisor of 1 + 2^4570 + 2^9832 + 2^16395