A341116 -id:A341116 - cp's OEIS frontend

A072288 Smallest prime factor of googolplex + n that exceeds 13, or 1 if googolplex + n is 13-smooth.

316912650057057350374175801344000001

Offset: 1

Ed Pegg Jr and Robert G. Wilson v, Nov 21 2002

a(2) is not known. After a(1) the sequence continues ?, 1429, 1129, 29, ?, 53, 427169, 5501, 19, 59, 1327, 1645318771, 61, 211, 17, 1831, ?, 43, 389, 173, 233886337, 139, 1451, 18797, 31, 37, 8297, 19, 13879, ?, 9241, 17, 29, ...

The question marks indicate terms > 10^14. - Dario Alpern, May 17 2003

Tanya Khovanova, Non Recursions
Dario Alejandro Alpern, Factors of 1000 numbers starting from googolplex
Robert Harley, Factors of Googolplex+1 (Found some terms)
Eric Weisstein's World of Mathematics, Googolplex
Index entries for one-term sequences

Cf. A067007, A078814, A341116.

(* For any individual n *) k = 17; While[ !PrimeQ[k] || PowerMod[10, 10^100, k] + n != k, k += 2]; k

Name edited by Peter Munn, Feb 20 2025

A341115 Numbers k such that k*2^101 + 1 is a prime factor of 10^(10^100) + 1.

Original entry on oeis.org

125000, 61298400, 578869250, 4511718750, 195312500000, 2918554687500, 3874552343750

Offset: 1

Yan Sheng Ang, Feb 05 2021

Every prime factor of 10^(10^100) + 1 is of the given form (k == 1 (mod 2^101)).
If k is not divisible by 10, then k == 1,3,4 (mod 10), and k*2^101 + 1 divides 10^(2^100) + 1.
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.
No other terms below 4*10^12. Other known terms in this sequence are 397299146187500, 194585800170898437500, 3163315773010253906250, 3274180926382541656494140625000, 128238752949982881546020507812500, 13940493204245285596698522567749023437500, 61902333925445418572053313255310058593750, 146251500493521646717454132158309221267700195312500.

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

D. A. Alpern, Factors of 1000 numbers starting from googolplex

Cf. A341116 (corresponding primes), A072288.

A341115_list, k, m, l, n = [], 1, 2**101, 2**101+1, 10**100
while k < 10**6:
    if pow(10,n,l) == l-1:
        A341115_list.append(k)
        print(len(A341115_list),k)
    k += 1
    l += m # Chai Wah Wu, Mar 28 2021

cp's OEIS Frontend

A072288 Smallest prime factor of googolplex + n that exceeds 13, or 1 if googolplex + n is 13-smooth.

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