A172290 -id:A172290 - cp's OEIS frontend

A242715 Prime divisors of 2^3510-1, listed with multiplicities.

3, 3, 3, 3, 7, 11, 19, 31, 73, 79, 131, 151, 271, 331, 631, 811, 937, 1171, 2731, 3511, 3511, 6553, 8191, 10531, 15121, 23311, 65521, 86113, 87211, 107251, 121369, 262657, 348031, 409891, 446473, 1024921, 1969111, 4633201, 7623851, 18837001, 22366891, 29121769

Offset: 1

Felix Fröhlich, May 21 2014

Amiram Eldar, Table of n, a(n) for n = 1..71
factordb, Factors of 2^3510-1.
Gerard P. Michon, Wieferich Primes 1093 and 3511

The corresponding sequence for 1093 is A172290.

Cf. A001220.

forprime(n=2, 1e6, m=n; while(Mod(2,m)^3510-1==0, print1(n, ", "); m=m*n))

More terms from Amiram Eldar, Oct 07 2019

A177855 Divisors of 2^1092 - 1.

Original entry on oeis.org

1, 3, 5, 7, 9, 13, 15, 21, 29, 35, 39, 43, 45, 49, 53, 63, 65, 79, 87, 91, 105, 113, 117, 127, 129, 145, 147, 157, 159, 169, 195, 203, 215, 237, 245, 261, 265, 273, 301, 313, 315, 337, 339, 371, 377, 381, 387, 395, 435, 441, 455, 471, 477, 507, 547, 553, 559, 565

Offset: 1

Reinhard Zumkeller, May 14 2010

The sequence is finite with A000005(2^1092-1) = 178120883699712 terms.

Chai Wah Wu, Table of n, a(n) for n = 1..10000 (Terms for n = 1..5000 from Reinhard Zumkeller)
Dario A. Alpern, Factorization using the Elliptic Curve Method
Index entries for sequences related to divisors of numbers

Cf. A000005, A172290, A172291, A046801.

divisors(2^1092-1) \\ Charles R Greathouse IV, Jun 21 2017

A177855_list = [i for i in range(1,10**6) if (2**1092-1) % i == 0] # Chai Wah Wu, Jun 02 2021

A172291 Numbers greater than 1 whose square divide 2^1092-1.

Original entry on oeis.org

3, 7, 13, 1093

Offset: 1

Artur Jasinski, Jan 30 2010

Up to now only two primes p such that p^2 divides 2^(p-1)-1 are known (these two are Wieferich primes, see A001220).

			From _Reinhard Zumkeller_, May 14 2010: (Start)
a(1)^2 = 9 = A177855(5);
a(2)^2 = 49 = A177855(14);
a(3)^2 = 169 = A177855(30);
a(4)^2 = 1194649 = A177855(1843). (End)

Cf. A001220, A172290.

Keyword full added by Reinhard Zumkeller, May 14 2010

A273085 Prime divisors of 68^112 - 1, listed with multiplicities.

Original entry on oeis.org

3, 5, 5, 5, 23, 29, 37, 41, 67, 113, 113, 113, 197, 617, 881, 10193, 103867, 521497, 938071, 1106356357, 1546157677, 100343116693, 518914006417, 1145565031404704513, 135178919999357237393881, 620712448371732926474772025689944913040651041015217889164158638163856301549281

Offset: 1

Felix Fröhlich, May 14 2016

(68^112-1)/113 is the only known Fermat quotient q_p(b) = (b^(p-1)-1)/p with 1 < b < p and q_p(b) divisible by p^2.

			68^112 == 1 (mod 113^3), but 68^112 =/= 1 (mod 113^4), so 113 appears three times in the sequence.

G. P. Michon, Prime Factorization of 68^112-1.

Cf. A172290, A242715.

forprime(p=1, 68^112-1, my(k=1); while(Mod(68, p^k)^112==1, print1(p, ", "); k++))

A343763 Common prime factors of A007663(183)/1093 and A007663(490)/3511.

Original entry on oeis.org

3, 7, 79, 2731, 8191, 121369, 22366891

Offset: 1

Felix Fröhlich, Apr 28 2021

The sequence is complete (cf. Michon).
Are these primes necessarily prime factors of A007663(i)/p when p = prime(i) is a Wieferich prime (A001220)?
Note the following curious observation: 3 * 7 * 79 * 2731 * 8191 * 121369 * 22366891 = 100743818301219097892181. That number is a repdigit in bases 4 and 64 and undulating in bases 2, 8, 128, 2048 and 8192. Compare that to observations from John Blythe Dobson (cf. Dobson).

J. B. Dobson, A note on the two known Wieferich primes
Gerard P. Michon, Wieferich Primes 1093 and 3511

Cf. A001220, A007663, A010785, A033619, A172290, A242715.

fq(n) = (2^(n-1)-1)/n
my(x=fq(1093)/1093, y=fq(3511)/3511); forprime(p=1, , if(Mod(x, p)==0 && Mod(y, p)==0, print1(p, ", ")))

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A242715 Prime divisors of 2^3510-1, listed with multiplicities.

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A177855 Divisors of 2^1092 - 1.

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A172291 Numbers greater than 1 whose square divide 2^1092-1.

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A273085 Prime divisors of 68^112 - 1, listed with multiplicities.

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A343763 Common prime factors of A007663(183)/1093 and A007663(490)/3511.

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PARI