A341115 - OEIS

%I #17 Mar 28 2021 15:54:24

%S 125000,61298400,578869250,4511718750,195312500000,2918554687500,

%T 3874552343750

%N Numbers k such that k*2^101 + 1 is a prime factor of 10^(10^100) + 1.

%C Every prime factor of 10^(10^100) + 1 is of the given form (k == 1 (mod 2^101)).

%C If k is not divisible by 10, then k == 1,3,4 (mod 10), and k*2^101 + 1 divides 10^(2^100) + 1.

%C If 1 <= j <= 99 and k is not divisible by 5^(j+1), then k*2^101 + 1 divides 10^(2^100*5^j) + 1.

%C No other terms below 4*10^12. Other known terms in this sequence are 397299146187500, 194585800170898437500, 3163315773010253906250, 3274180926382541656494140625000, 128238752949982881546020507812500, 13940493204245285596698522567749023437500, 61902333925445418572053313255310058593750, 146251500493521646717454132158309221267700195312500.

%H D. A. Alpern, <a href="https://www.alpertron.com.ar/GOOGOL.HTM">Factors of 1000 numbers starting from googolplex</a>

%e The smallest prime factor of 10^10^100 + 1 is 125000*2^100 + 1 = 316912650057057350374175801344000001.

%o (Python)

%o A341115_list, k, m, l, n = [], 1, 2**101, 2**101+1, 10**100

%o while k < 10**6:

%o     if pow(10,n,l) == l-1:

%o         A341115_list.append(k)

%o         print(len(A341115_list),k)

%o     k += 1

%o     l += m # _Chai Wah Wu_, Mar 28 2021

%Y Cf. A341116 (corresponding primes), A072288.

%K nonn,fini,hard,more

%O 1,1

%A _Yan Sheng Ang_, Feb 05 2021