A359650
Smallest prime factor q of (2^(p-1)-1) / (3*p) with prime p such that q is greater than p (increasing p, cf. A359387).
Original entry on oeis.org
31, 89, 178481, 233, 13367, 6361, 499, 62020897, 3391, 1049, 4153, 1433, 7068569257, 1327, 1399, 1913, 54217, 80929, 26371, 7753, 855857, 5867, 3449, 48731, 7707719, 12619129, 104369, 32051, 78557207, 67219, 1676083, 34513, 22291, 4567, 14563, 830833, 2731, 343081
Offset: 1
For p=7, (2^6-1)/(3*7) = 3 and 3 is not greater than 7.
For p=11, (2^10-1)/(3*11) = 31, which is greater than 11, so a(1)=31.
For p=13, (2^12-1)/(3*13) = 105 = 3*5*7 and 3 is not greater than 13.
For p=17, (2^16-1)/(3*17) = 1285 = 5*257 and 5 is not greater than 17.
For p=19, (2^18-1)/(3*19) = 4599 = 3^2*7*73 and 3 is not greater than 19.
For p=23, (2^22-1)/(3*23) = 60787 = 89*683 and 89 is greater than 23, so a(2)=89.
A360827
Primes p, not safe primes, such that the smallest factor of (2^(p-1)-1) / 3 is equal to p.
Original entry on oeis.org
443, 647, 1847, 2243, 2687, 2699, 6263, 6563, 7487, 7583, 8627, 8663, 9419, 9767, 10223, 11867, 12323, 13187, 13907, 14627, 14723, 14783, 17747, 17783, 19739, 20639, 20807, 21863, 22307, 23747, 24107, 24923, 25127, 26759, 27983, 29207, 29819, 30839, 31247, 32303, 34403, 34439
Offset: 1
443 is the first term since p = 443 is the first term of A359387 that is not in A005385 (i.e., (443-1)/2 = 13*17 is not prime).
647 is the second term since p = 647 is the first term (> 443) of A359387 that is not in A005385 (i.e., (647-1)/2 = 17*19 is not prime).
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q[p_] := ! PrimeQ[(p - 1)/2] && AllTrue[Range[p], ! PrimeQ[#] || PowerMod[2, p - 1, 3*p*#] > 1 &]; Select[Prime[Range[4, 4000]], q] (* Amiram Eldar, Mar 01 2023 *)
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forprime(p=11, 40000, if(!isprime((p-1)/2), forprime(div=5, p-1, if(Mod(2, div)^(p-1)==1, next(2))); print1(p, ", ")))
Showing 1-2 of 2 results.