A077816 -id:A077816 - cp's OEIS frontend

A253016 Numbers k such that 11^phi(k) == 1 (mod k^2), where phi(k) = A000010(k).

71, 142, 284, 355, 497, 710, 994, 1420, 1491, 1988, 2485, 2840, 2982, 3976, 4970, 5680, 5964, 7455, 9940, 11928, 14910, 19880, 23856, 29820, 39760, 59640, 79520, 119280, 238560, 477120

Offset: 1

Felix Fröhlich, Dec 26 2014

nonn

No further terms up to 10^9.
No more terms less than 10^10. - Robert G. Wilson v, Jan 18 2015
The first 30 terms are divisible by 71. Are there any terms not divisible by 71? - Robert Israel, Dec 30 2014
By Corollary 5.9 in Agoh, Dilcher, Skula (1997), if there are no further Wieferich primes to base 11 apart from 71, then the answer is no. - Felix Fröhlich, Dec 30 2014

T. Agoh, K. Dilcher and L. Skula, Fermat Quotients for Composite Moduli, J. Num. Theory, Vol. 66, Issue 1 (1997), 29-50.

Cf. A077816, A242958, A242959, A242960, A245529, A241977, A241978.

select(t -> 11 &^ numtheory:-phi(t) mod t^2 = 1, [$1..10^6]); # Robert Israel, Dec 30 2014

a253016[n_] := Select[Range[n], PowerMod[11,EulerPhi[#], #^2] == 1 &]; a253016[500000] (* Michael De Vlieger, Dec 29 2014; modified by Robert G. Wilson v, Jan 18 2015 *)

for(n=2, 1e9, if(Mod(11, n^2)^(eulerphi(n))==1, print1(n, ", ")))

A237292 a(n) = A002326(2n(n+1)) / A002326(n).

Original entry on oeis.org

1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 7, 23, 25, 27, 29, 31, 33, 35, 37, 13, 41, 43, 45, 47, 49, 51, 53, 11, 19, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99, 101, 103, 35, 107, 109, 37, 113, 115, 117, 119, 121, 123, 125

Offset: 0

Thomas Ordowski, Feb 06 2014

nonn

Note that ((2n+1)^2-1)/2 = 2n(n+1).
We have 1 <= a(n) <= 2n+1 and a(n) divides 2n+1 for every n >= 0.
Odd m is a Wieferich number A182297 if and only if a((m-1)/2) < m.
Odd prime p is a Wieferich prime A001220 if and only if a((p-1)/2) = 1.
a((n-1)/2) = 1 for n = 1, 1093, 3511, 7651, 10533, 14209, 17555, ...

Robert Israel, Table of n, a(n) for n = 0..10000

Cf. A001220, A002326, A077816 and A182297.

1,seq(numtheory:-order(2,4*n*(n+1)+1)/numtheory:-order(2,2*n+1),n=1..100); # Robert Israel, Dec 02 2015

a002326(n) = znorder(Mod(2, 2*n+1));
a(n) = a002326(2*n*(n+1))/a002326(n); \\ Michel Marcus, Feb 08 2014

a(n) = ord_{(2n+1)^2}(2) / ord_{2n+1}(2), n >= 0.

More terms from Michel Marcus, Feb 08 2014

Edited by Thomas Ordowski, Dec 02 2015

A218584 Numbers n such that 2^A000010(nx) == 1 (mod (nx)^2), where x = 3837523 is the product of the first 2 Wieferich primes 1093 and 3511.

Original entry on oeis.org

1, 3, 5, 7, 9, 13, 15, 21, 27, 35, 39, 45, 63, 65, 81, 91, 105, 117, 135, 169, 189, 195, 273, 315, 351, 405, 455, 507, 567, 585, 819, 845, 945, 1053, 1183, 1365, 1521, 1701, 1755, 2457, 2535, 2835, 3159, 3549, 4095, 4563, 5265, 5915, 7371, 7605, 8505, 10647

Offset: 1

Arkadiusz Wesolowski, Nov 03 2012

nonn

3837523*a(n) is a term in A077816.

If this sequence is finite, then there are finitely many Wieferich primes (A001220).

Arkadiusz Wesolowski, Table of n, a(n) for n = 1..74

Cf. A001220, A077816.

x = 3837523; Select[Range[1, 10647, 2], PowerMod[2, EulerPhi[#*x], (#*x)^2] == 1 &]

is(n)=Mod(2,(3837523*n)^2)^eulerphi(3837523*n)==1 \\ Charles R Greathouse IV, Mar 05 2013

A247154 a(n) = smallest composite c such that n^(A000010(c)) == 1 (mod c^2), i.e., smallest composite Wieferich number to base n.

Original entry on oeis.org

4, 3279, 22, 3279, 41542, 330805, 4, 3279, 4, 1461, 142, 1812389, 1726, 3883, 4, 3279, 4, 35, 6, 1967

Offset: 1

Felix Fröhlich, Nov 21 2014

a(21) > 10^9
a(22)-a(28): 39, 4, 128165, 4, 9, 22, 9
a(29) > 10^9
a(30)-a(33): 1123787, 4, 3279, 4
a(34) > 10^9

William D. Banks, Florian Luca, Igor E. Shparlinski, Estimates for Wieferich numbers, The Ramanujan Journal, December 2007, Volume 14, Issue 3, pp 361-378.
Takashi Agoh, Karl Dilcher, Ladislav Skula, Fermat Quotients for Composite Moduli, Journal of Number Theory, September 1997, Volume 66, Issue 1, pp 29-50.

Cf. A039951, A077816, A241977, A241978, A242958, A242959, A242960, A245529.

for(n=1, 20, forcomposite(c=1, 1e9, if(Mod(n, c^2)^(eulerphi(c))==1, print1(c, ", "); next({2}))); print1("--, "))

A250206 Least base b > 1 such that b^A000010(n) = 1 (mod n^2).

Original entry on oeis.org

2, 5, 8, 7, 7, 17, 18, 15, 26, 7, 3, 17, 19, 19, 26, 31, 38, 53, 28, 7, 19, 3, 28, 17, 57, 19, 80, 19, 14, 107, 115, 63, 118, 65, 18, 53, 18, 69, 19, 7, 51, 19, 19, 3, 26, 63, 53, 17, 18, 57, 134, 19, 338, 161, 3, 31, 28, 41, 53, 107, 264, 115, 19, 127, 99, 161, 143, 65, 28, 99, 11, 55

Offset: 1

Eric Chen, Feb 21 2015

nonn

a(n) = least base b > 1 such that n is a Wieferich number (see A077816).
At least, b = n^2+1 can satisfy this equation, so a(n) is defined for all n.
Least Wieferich number (>1) to base n: 2, 1093, 11, 1093, 2, 66161, 4, 3, 2, 3, 71, 2693, 2, 29, 4, 1093, 2, 5, 3, 281, 2, 13, 4, 5, 2, ...; each is a prime or 4. It is 4 if and only if n mod 72 is in the set {7, 15, 23, 31, 39, 47, 63}.
Does every natural number (>1) appear in this sequence? If yes, do they appear infinitely many times?
For prime n, a(n) = A185103(n), does there exist any composite n such that a(n) = A185103(n)?

			a(30) = 107 since A000010(30) = 8, 30^2 = 900, and 107 is the least base b > 1 such that b^8 = 1 (mod 900).

Eric Chen, Table of n, a(n) for n = 1..1000

Cf. A039678, A185103, A125636, A039951, A247154, A001220, A014127, A123692, A212583, A123693, A045616, A111027, A128667, A234810, A242741, A128668, A244260, A090968, A242982, A128669, A077816, A242958, A242959, A241978, A242960, A241977, A253016, A245529.

f[n_] := Block[{b = 2, m = EulerPhi[n]}, While[ PowerMod[b, m, n^2] != 1, b++]; b]; f[1] = 2; Array[f, 72] (* Robert G. Wilson v, Feb 28 2015 *)

a(n)=for(k=2,2^24,if((k^eulerphi(n))%(n^2)==1, return(k)))

a(prime(n)) = A039678(n) = A185103(prime(n)).
a(A077816(n)) = 2.
a(A242958(n)) <= 3.

A257660 Numbers n such that 13^phi(n) == 1 (mod n^2), where phi(n) = A000010(n).

Original entry on oeis.org

2, 863, 1726, 3452, 371953, 743906, 1487812, 1747591, 1859765, 2975624, 3495182, 3719530, 5242773, 6990364, 7439060, 8737955, 10485546, 14878120, 15993979, 17475910, 20971092, 26213865, 29756240, 31987958, 34951820, 41942184, 47981937, 52427730, 59512480

Offset: 1

Felix Fröhlich, Jul 26 2015

nonn

The subsequence of primes in this sequence is A128667.

Giovanni Resta, Table of n, a(n) for n = 1..140 (terms < 3*10^11)

Cf. A077816, A128667, A241977, A241978, A242958, A242959, A242960, A245529, A253016.

Select[Range@ 1000000, Mod[13^EulerPhi[#], #^2] == 1 &] (* Michael De Vlieger, Jul 27 2015 *)

for(n=2, 1e9, if(Mod(13, n^2)^(eulerphi(n))==1, print1(n, ", ")))

A265630 Numbers n > 1 such that 2^m == 1 (mod n^2), where m = A002326((n-1)/2).

Original entry on oeis.org

1093, 3511, 7651, 10533, 14209, 17555, 22953, 31599, 42627, 45643, 52665, 99463, 136929, 157995, 228215, 298389, 410787, 684645, 2053935, 3837523, 11512569, 19187615, 26862661, 34537707, 49887799, 57562845, 80587983, 134313305, 149663397, 172688535, 241763949, 249438995, 349214593, 402939915, 448990191, 748316985, 1047643779, 1208819745, 1746072965, 2244950955, 3142931337, 5238218895, 15714656685

Offset: 1

Thomas Ordowski, Dec 11 2015

nonn

A subsequence of A077816.
Odd numbers n > 1 such that A237292((n-1)/2) = 1.
Indices k such that A077816(k) is not in this sequence are 2, 13, 14, 16, 22, 24, 25, 27, 28, 30, ... - Altug Alkan, Dec 11 2015
There are no other terms, unless there are other Wieferich primes (A001220) besides 1093 and 3511. - Max Alekseyev, Dec 11 2015

Cf. A002326, A077816, A237292.

Select[Range[10^6], Mod[2^MultiplicativeOrder[2, 2 ((# - 1)/2) + 1], #^2] == 1 &] (* Michael De Vlieger, Dec 11 2015 *)

a002326(n) = znorder(Mod(2, 2*n+1));
a237292(n) = a002326(2*n*(n+1))/a002326(n);
for(n=1, 1e8, if(a237292(n)==1, print1(2*n+1, ", "))) \\ Altug Alkan, Dec 11 2015

More terms from Altug Alkan, Dec 11 2015

Missing terms a(28)-a(30) and further terms a(34)-a(43) added by Max Alekseyev, Dec 11 2015

A317475 Numbers k such that k^2 | A038199(k).

Original entry on oeis.org

1, 16, 32, 64, 112, 128, 256, 395, 448, 512, 1024, 1093, 1168, 1368, 1472, 1792, 2013, 2048, 3279, 3344, 3511, 3968, 4096, 5472, 5696, 7168, 7651, 8192, 10533, 14209, 16384, 17488, 19674, 21672, 21888, 22953, 27552, 28672, 31599, 32768, 33883, 34905, 34976

Offset: 1

Amiram Eldar, Jul 29 2018

nonn

Serret proved in 1855 a generalization of Fermat's little theorem: for b >= 1, Sum_{d|k} mu(d)*b^(k/d) == 0 (mod k). This sequence includes numbers k such that k^2 divides the sum with base b=2.
Includes all the powers of 2 above 8.
An alternative generalization of Wieferich primes (A001220) which are the prime terms of this sequence.
Also numbers k such that k | A059966(k).

			16 is in the sequence since Sum_{d|16} mu(d)*2^(16/d) = 65280 = 255 * 16^2.

Wacław Sierpiński, Elementary Theory of Numbers, Elsevier, North Holland, 1988, page 217.

Amiram Eldar, Table of n, a(n) for n = 1..250
Joseph-Alfred Serret, Théorème de Fermat généralisé, Nouvelles Annales de Mathématiques, Vol. 14 (1855), pp. 261-262.

Cf. A000079, A001220, A038199, A059966, A077816, A182297.

f[n_] := DivisorSum[n, MoebiusMu[#] * 2^(n/#) &]; Select[Range[1000], Divisible[f[#], #^2] &]

isok(n) = frac(sumdiv(n, d, moebius(n/d)*(2^d-1))/n^2) == 0; \\ Michel Marcus, Jul 30 2018

A291961 Numbers n > 1 such that 2^lambda(n) == 1 (mod n^2), where lambda(n) is the Carmichael lambda function (A002322).

Original entry on oeis.org

1093, 3279, 3511, 7651, 10533, 14209, 17555, 22953, 31599, 42627, 45643, 52665, 99463, 136929, 157995, 228215, 298389, 410787, 684645, 2053935, 3837523, 11512569, 19187615, 26862661, 34537707, 49887799, 57562845, 80587983, 134313305, 149663397, 172688535, 241763949, 249438995, 349214593, 402939915, 448990191, 748316985, 1047643779, 1208819745, 1746072965, 2244950955, 3142931337, 5238218895

Offset: 1

Amiram Eldar, Sep 06 2017

nonn

An alternative generalization of Wieferich primes (A001220).
A subsequence of A077816, since the A002322(n)|A000010(n). The first 12 terms are common.
15714656685 (see A265630) is also a term. - Michel Marcus, Sep 14 2017

Cf. A265630, A001220, A002322, A077816, A182297, A246503.

Select[Range[2, 100000], Divisible[2^CarmichaelLambda[#] - 1, #^2] &]

isok(n) = Mod(2, n^2)^lcm(znstar(n)[2]) == 1; \\ Michel Marcus, Sep 11 2017

a(32)-a(36) from Michel Marcus, Sep 11 2017
a(37)-a(41) from Michel Marcus, Sep 12 2017
a(42)-a(43) from Michel Marcus, Sep 14 2017

A319314 Numbers k such that 2^phi(k) == phi(k)^2 (mod k^2).

Original entry on oeis.org

1, 3, 4, 5, 6, 8, 10, 12, 384, 640, 768, 896, 960, 24576, 49152, 950272, 1425408, 1572864, 3145728, 10485760, 19398656, 65011712, 100663296, 110057537, 201326592, 220115074, 671088640, 1879048192

Offset: 1

Altug Alkan, Sep 17 2018

Sequence is infinite, i.e., 3*2^(3*(t-1)-(-1)^t) is a term for all t > 0.

Prime terms (5, 110057537, ...) are in A246568 based on case A = +1.

Cf. A077815, A077816, A246568, A292544.

[1] cat [n: n in [1..10^6] | 2^EulerPhi(n) mod n^2 eq EulerPhi(n)^2]; // Vincenzo Librandi, Sep 20 2018

isok(n) = Mod(2, n^2)^eulerphi(n)==eulerphi(n)^2;

cp's OEIS Frontend

A253016 Numbers k such that 11^phi(k) == 1 (mod k^2), where phi(k) = A000010(k).

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A237292 a(n) = A002326(2n(n+1)) / A002326(n).

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A218584 Numbers n such that 2^A000010(n*x) == 1 (mod (n*x)^2), where x = 3837523 is the product of the first 2 Wieferich primes 1093 and 3511.

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A247154 a(n) = smallest composite c such that n^(A000010(c)) == 1 (mod c^2), i.e., smallest composite Wieferich number to base n.

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A250206 Least base b > 1 such that b^A000010(n) = 1 (mod n^2).

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A257660 Numbers n such that 13^phi(n) == 1 (mod n^2), where phi(n) = A000010(n).

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A265630 Numbers n > 1 such that 2^m == 1 (mod n^2), where m = A002326((n-1)/2).

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A317475 Numbers k such that k^2 | A038199(k).

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A218584 Numbers n such that 2^A000010(nx) == 1 (mod (nx)^2), where x = 3837523 is the product of the first 2 Wieferich primes 1093 and 3511.