A243905 -id:A243905 - cp's OEIS frontend

A258367 a(n) is the smallest A (in absolute value) such that for p = prime(n), 2^{(p-1)/2} == +-1 + A*p (mod p^2), i.e., such that p is a near-Wieferich prime.

1, 1, 1, 3, 5, 2, 8, 3, 14, 3, 18, 9, 9, 22, 18, 4, 18, 5, 1, 28, 30, 24, 3, 20, 46, 22, 47, 21, 15, 9, 57, 42, 15, 48, 28, 41, 48, 60, 85, 25, 74, 25, 52, 11, 32, 51, 17, 13, 34, 113, 13, 71, 2, 16, 64, 130, 81, 35, 37, 29, 39, 147, 68, 60, 71, 96, 92, 99, 12

Offset: 2

Felix Fröhlich, May 28 2015

nonn

p is in A001220 iff a(n) = 0. This is the case iff A014664(n) = A243905(n), which happens for n = 183 and n = 490.

Is a(n) = 0 for any other n, and, if yes, are there infinitely many such n?

Felix Fröhlich, Table of n, a(n) for n = 2..10000
R. Crandall, K. Dilcher and C. Pomerance, A search for Wieferich and Wilson primes, Mathematics of Computation, 66 (1997), 433-449.

Cf. A001220, A195988, A241014, A244801, A246568, A250406, A250407, A306885, A338646, A343410.

a(n,p=prime(n))=abs(centerlift(Mod(2,p^2)^((p-1)/2))\/p)
apply(p->a(0,p), primes(100)[2..100]) \\ Charles R Greathouse IV, Jun 15 2015

a(n) = min(b(n) mod p, -b(n) mod p) where p = prime(n) and b(n) = Sum_{i=1..ceiling((p-1)/4)} (2i-1)^(p-2). - Daniel Chen, Sep 01 2022

A282902 Square array A(n, k) read by antidiagonals downwards: multiplicative order of 2 modulo prime(n)^k, where k runs over the positive integers.

Original entry on oeis.org

2, 6, 4, 18, 20, 3, 54, 100, 21, 10, 162, 500, 147, 110, 12, 486, 2500, 1029, 1210, 156, 8, 1458, 12500, 7203, 13310, 2028, 136, 18, 4374, 62500, 50421, 146410, 26364, 2312, 342, 11, 13122, 312500, 352947, 1610510, 342732, 39304, 6498, 253, 28, 39366, 1562500, 2470629, 17715610, 4455516, 668168, 123462, 5819, 812, 5

Offset: 2

Felix Fröhlich, Feb 24 2017

The number of initial terms in row n with constant values is equal to the highest value of x such that p = prime(n) satisfies 2^(p-1) == 1 (mod p^x).
From Robert Israel, Feb 24 2017: (Start)
a(n,k+1) is either a(n,k) or a(n,k)*prime(n). If it is a(n,k)*prime(n), then a(n,k+j) = a(n,k)*prime(n)^j for all j>=1.
a(n,2) = a(n,1) if and only if prime(n) is a Wieferich prime (A001220).
(End)

			Array A(n, k) starts
   2,   6,   18,     54,     162,      486,      1458
   4,  20,  100,    500,    2500,    12500,     62500
   3,  21,  147,   1029,    7203,    50421,    352947
  10, 110, 1210,  13310,  146410,  1610510,  17715610
  12, 156, 2028,  26364,  342732,  4455516,  57921708
   8, 136, 2312,  39304,  668168, 11358856, 193100552
  18, 342, 6498, 123462, 2345778, 44569782, 846825858

Robert Israel, Table of n, a(n) for n = 2..10012 (first 142 antidiagonals, flattened)

Cf. A014664 (column 1), A243905 (column 2).

Cf. A001220.

seq(seq(numtheory:-order(2,ithprime(i)^(m-i)),i=2..m-1),m=2..10); # Robert Israel, Feb 24 2017

A[n_, k_] := MultiplicativeOrder[2, Prime[n]^k];
Table[A[n-k+1, k], {n, 2, 11}, {k, n-1, 1, -1}] // Flatten (* Jean-François Alcover, Mar 02 2020 *)

a(n, k) = znorder(Mod(2, prime(n)^k))
array(rows, cols) = for(n=2, rows+1, for(k=1, cols, print1(a(n, k), ", ")); print(""))
array(7, 8) \\ print 7 X 8 array

A187965 Numbers k such that 2^k - 1 and 2^k + 1 are not squarefree.

Original entry on oeis.org

21, 30, 63, 78, 90, 105, 110, 147, 150, 189, 204, 210, 231, 234, 270, 273, 310, 315, 330, 340, 357, 390, 399, 441, 450, 465, 483, 510, 525, 546, 550, 567, 570, 609, 612, 630, 651, 657, 666, 690, 693, 702, 735, 750, 759, 770, 777, 810, 819, 858, 861, 870, 903, 930, 945, 987, 990, 1014, 1020, 1029, 1050, 1071

Offset: 1

Vladimir Joseph Stephan Orlovsky, Mar 18 2011

nonn

If k is in the sequence, then so is m*k for any odd m. - Thomas Ordowski, Nov 23 2015
Note that 110, 310, 340, 550, 770 are not divisible by 3.
Let b(p) be the multiplicative order of 2 modulo p^2. Then k is in this sequence if and only if there exists odd primes p, q such that b(p) | k and k == b(q)/2 (mod b(q)) with even b(q). For example, we have b(7) = 21, b(3) = 6 so b(7) | 21, 21 == b(3)/2 (mod b(3)), hence 21 is a term; likewise, b(3) = 6, b(5) = 20, so b(3) | 30, 30 == b(5)/2 (mod b(5)), hence 30 is a term. - Jianing Song, Jan 20 2021

			2^21 - 1 = 7^2 * 127 * 337, 2^21 + 1 = 3^2 * 43 * 5419.

Cf. A005117, A049094, A049096.

Cf. A243905 (multiplicative orders of 2 modulo p^2), A242777 (k+1 is prime).

[n: n in [1..250] | not IsSquarefree(2^n-1) and not IsSquarefree(2^n+1)]; // Vincenzo Librandi, Nov 23 2015

Select[ Range@ 500, !(SquareFreeQ[2^# - 1] || SquareFreeQ[2^# + 1]) &]
Select[Range[1100],NoneTrue[2^#+{1,-1},SquareFreeQ]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jun 22 2019 *)

is(n) = !issquarefree(2^n-1) && !issquarefree(2^n+1);
for(n=1, 1e3, if(is(n), print1(n, ", "))) \\ Altug Alkan, Nov 22 2015

More terms from Joerg Arndt, Nov 23 2015

A282552 Difference between the multiplicative orders of 2 modulo p^2 and 2 modulo p, where p = prime(n).

Original entry on oeis.org

4, 16, 18, 100, 144, 128, 324, 242, 784, 150, 1296, 800, 588, 1058, 2704, 3364, 3600, 4356, 2450, 648, 3042, 6724, 968, 4608, 10000, 5202, 11236, 3888, 3136, 882, 16900, 9248, 19044, 21904, 2250, 8112, 26244, 13778, 29584, 31684, 32400, 18050, 18432, 38416

Offset: 2

Felix Fröhlich, Feb 18 2017

nonn

a(n) = 0 iff A014664(n) = A243905(n), i.e., iff prime(n) is a Wieferich prime (A001220). So far this is known to be the case only for prime(183) = 1093 and prime(490) = 3511, i.e., a(183) = 0 and a(490) = 0.

Cf. A001220, A014664, A243905.

Table[MultiplicativeOrder[2, #^2] - MultiplicativeOrder[2, #] &@ Prime@ n, {n, 2, 45}] (* Michael De Vlieger, Feb 18 2017 *)

a(n) = my(p=prime(n)); znorder(Mod(2, p^2)) - znorder(Mod(2, p))

a(n) = A243905(n) - A014664(n).

A305184 Multiplicative order of 2 (mod p^2), where p is the n-th Wieferich prime (A001220).

Original entry on oeis.org

364, 1755

Offset: 1

Felix Fröhlich, May 30 2018

Meissner discovered the congruence 2^364 == 1 (mod 1093^2) and thus proved that 1093 is a Wieferich prime, i.e., a term of A001220 (cf. Meissner, 1913).
Later, Beeger discovered the congruence 2^1755 == 1 (mod 3511^2) and proved that 3511 is also a Wieferich prime (cf. Beeger, 1922).
Let b(n) = (A001220(n)-1)/a(n). Then b(1) = 3 and b(2) = 2.
From the fact that a(1) and a(2) are composite it follows that A001220(1) = 1093 and A001220(2) = 3511 do not divide any terms of A001348 (cf. Dobson).
Curiously, both 364 and 1755 are repdigits in some base. 364 = 444 in base 9 and 1755 = 3333 in base 8. Compare this with Dobson's observation that 1092 and 3510 are 444 in base 16 and 6666 in base 8, respectively (cf. Dobson).

N. G. W. H. Beeger, On a new case of the congruence 2^p-1 == 1 (mod p^2), Messenger of Mathematics 51 (1922), 149-150.
J. B. Dobson, A note on the two known Wieferich primes
W. Meissner, Über die Teilbarkeit von 2^p-2 durch das Quadrat der Primzahl p = 1093, Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften Berlin, 35 (1913), 663-667. [Annotated scanned copy]

Cf. A001220, A001348, A014664, A243905, A282552, A282902.

forprime(p=1, , if(Mod(2, p^2)^(p-1)==1, print1(znorder(Mod(2, p^2)), ", ")))

a(n) = A014664(A000720(A001220(n))) = A243905(A000720(A001220(n))). [Corrected by Jianing Song, Sep 20 2019]

cp's OEIS Frontend

A258367 a(n) is the smallest A (in absolute value) such that for p = prime(n), 2^{(p-1)/2} == +-1 + A*p (mod p^2), i.e., such that p is a near-Wieferich prime.

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A282902 Square array A(n, k) read by antidiagonals downwards: multiplicative order of 2 modulo prime(n)^k, where k runs over the positive integers.

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A187965 Numbers k such that 2^k - 1 and 2^k + 1 are not squarefree.

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A282552 Difference between the multiplicative orders of 2 modulo p^2 and 2 modulo p, where p = prime(n).

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A305184 Multiplicative order of 2 (mod p^2), where p is the n-th Wieferich prime (A001220).

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