A072288 -id:A072288 - cp's OEIS frontend

A076848 Smallest prime factor of googol + n that exceeds 13, or 1 if googol + n is 13-smooth.

73, 4832936419, 157, 20794121, 127, 859493, 557, 113, 3221, 19, 7549, 43, 17, 61, 211, 241, 18617, 907, 419, 47, 443, 911, 47955653711170550856726386495271851, 1109, 53, 31, 37, 2543, 19, 17, 617, 33521, 23, 7906914473, 38851, 421, 39640576062095087

Offset: 1

Jason Earls, Nov 23 2002

nonn

A googol is 10^100. Factors found using PARI and Dario Alpern's ECM factorization applet.

Sean A. Irvine, Table of n, a(n) for n = 1..1000 (terms 1..400 from Charles R Greathouse IV, n=260 corrected by Sean A. Irvine)
Dario Alpern, Factorization using the Elliptic Curve Method

Cf. A049014 (n such that googol + n is prime), A066298, A080197 (relates to positions of 1's).
Equivalent sequences: A072288 (googolplex + n), A078813 (googol - n).
See the formula section for the relationships with A007947, A020639, A034386.

a(n) = A020639(A007947(10^100 + n)/gcd(10^100 + n, A034386(13))), where A020639(m) = lpf(m), smallest prime factor of m. - Peter Munn, Feb 20 2025

a(23) found by Sean A. Irvine on Dec 08 2002 by employing SNFS, using the polynomials x^5+23 and x-10^20.
Edited by Robert G. Wilson v, Dec 09 2002
Edited by Peter Munn, Feb 20 2025

A078814 Smallest prime factor of googolplex - n that exceeds 13, or 1 if googolplex - n is 13-smooth.

Original entry on oeis.org

17, 1433, 499679, 1279, 31, 149

Offset: 1

Ed Pegg Jr and Robert G. Wilson v, Dec 06 2002

Seventh term is not known. Sequence continues ?, 625392489737, 19, 37, 5419, 107, 23, 60149, 1733, 89, 8543, 17, 5261, 229, 8656871, 4273, 14009, 29, 509, 43628661784403, 24539783, 19, 347, 12414692011, 523, 151, 421, 35816135619181, 17, 23, 184309, ?, 181, 288481, 163, 41, 743, 13337, 71, 53, 19, 59, 113, 37957721, ... where the unknown numbers exceed 25 * 10^12. - Dario Alpern, Jul 03 2003

Dario Alejandro Alpern, Factors of 1000 numbers up to googolplex.

Cf. A072288, A200924, A227247.

(* For any individual n *) k = 17; While[ !PrimeQ[k] || PowerMod[10, 10^100, k] - n != 0, k += 2]; k

a(n) = forprime(p=17, oo, if(Mod(10, p)^lift(Mod(10, p-1)^100) == n, return(p))); \\ Jinyuan Wang, Apr 17 2020

Name edited by Peter Munn, Feb 20 2025

A341116 Prime factors of 10^(10^100) + 1.

Original entry on oeis.org

316912650057057350374175801344000001, 155409907106060194289411023528840396801, 1467607904432329964944690923937202176001, 11438565962996913740067907829760000000001, 495176015714152109959649689600000000000001, 7399415202816574979127045311692800000000001, 9823157208340761024963422324575436800000001

Offset: 1

Yan Sheng Ang, Feb 05 2021

Every term of this sequence is == 1 (mod 2^101); see A341115 for more properties.

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

Cf. A072288, A341115.

a(n) = 1 + A341115(n)*2^101. - Jinyuan Wang, Feb 09 2021

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

A076848 Smallest prime factor of googol + n that exceeds 13, or 1 if googol + n is 13-smooth.

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

Extensions

A078814 Smallest prime factor of googolplex - n that exceeds 13, or 1 if googolplex - n is 13-smooth.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A341116 Prime factors of 10^(10^100) + 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Python