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A341115 Numbers k such that k*2^101 + 1 is a prime factor of 10^(10^100) + 1. 1
125000, 61298400, 578869250, 4511718750, 195312500000, 2918554687500, 3874552343750 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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.

LINKS

Table of n, a(n) for n=1..7.

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

EXAMPLE

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

PROG

(Python)

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

CROSSREFS

Cf. A341116 (corresponding primes), A072288.

Sequence in context: A182658 A282919 A100406 * A183797 A234783 A206134

Adjacent sequences:  A341112 A341113 A341114 * A341116 A341117 A341118

KEYWORD

nonn,fini,hard,more

AUTHOR

Yan Sheng Ang, Feb 05 2021

STATUS

approved

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Last modified October 17 21:17 EDT 2021. Contains 348065 sequences. (Running on oeis4.)