A242960 -id:A242960 - cp's OEIS frontend

A242959 Numbers n such that 5^A000010(n) == 1 (mod n^2).

2, 20771, 40487, 41542, 80974, 83084, 161948, 643901, 1255097, 1287802, 1391657, 1931703, 2510194, 2575604, 2783314, 3765291, 3863406, 4174971, 5020388, 5151208, 5566628, 7530582, 7726812, 8349942, 10040776, 11133256, 15061164, 15308227, 15453624, 16699884

Offset: 1

Felix Fröhlich, May 27 2014

nonn

Giovanni Resta, Table of n, a(n) for n = 1..479 (terms < 10^15, first 75 terms from Felix Fröhlich)

Cf. A077816, A242958, A242960.

If a(n) is prime, then a(n) is in A123692.

Select[Range[167*10^5],PowerMod[5,EulerPhi[#],#^2]==1&] (* Harvey P. Dale, Jun 02 2020 *)

for(n=2, 10^9, if(Mod(5, n^2)^(eulerphi(n))==1, print1(n, ", ")))

A242958 Numbers n such that 3^phi(n) == 1 (mod n^2), where phi(n) is Euler's totient function.

Original entry on oeis.org

11, 22, 44, 55, 110, 220, 440, 880, 1006003, 2012006, 4024012, 11066033, 22132066, 44264132, 55330165, 88528264, 110660330, 221320660, 442641320, 885282640, 1770565280, 56224501667, 112449003334, 224898006668, 393571511669, 449796013336, 618469518337, 787143023338

Offset: 1

Felix Fröhlich, May 27 2014

a(21) > 10^9.

All listed composite terms are multiples of the two known primes in this sequence, 11 and 1006003, the only known base 3 Wieferich primes.

Giovanni Resta, Table of n, a(n) for n = 1..78 (terms < 10^15)

Cf. A014127, A077816, A242959, A242960, A000010.

Select[Range[1000], Mod[3^EulerPhi[#], #^2] == 1 &] (* Alonso del Arte, Jun 02 2014 *)

for(n=2, 10^9, if(Mod(3, n^2)^(eulerphi(n))==1, print1(n, ", ")))

Terms a(21) and beyond from Giovanni Resta, Jan 27 2020

A245529 Numbers n such that 12^phi(n) == 1 (mod n^2), where phi(n) is Euler's totient function.

Original entry on oeis.org

2693, 123653, 1812389, 2349407, 12686723, 201183431, 332997529, 3822485189, 6326953051, 54520709801, 224107337017, 272603549005, 541786979683, 1035893486219, 1568751359119, 4258039403323, 5179467431095, 10293952613977, 29806275823261

Offset: 1

Felix Fröhlich, Jul 25 2014

a(8) > 10^9.
If a(n) is prime, it is in A111027.
a(20) > 10^14. - Giovanni Resta, Jan 27 2020

Takashi Agoh, Karl Dilcher and Ladislav Skula, Fermat Quotients for Composite Moduli, J. Number Theory, Volume 66, Issue 1 (1997), 29-50.

Cf. A000010, A077816, A242958, A242959, A242960, A111027.

with(numtheory): A245529:=n->`if`( (12 &^ phi(n)-1) mod n^2 = 0, n, NULL): seq(A245529(n), n=2..10^4); # Wesley Ivan Hurt, Jul 26 2014

Select[Range[10^5], PowerMod[12, EulerPhi[#], #^2] == 1 &] (* Alonso del Arte, Jul 27 2014 *)

for(n=2, 1e9, if(Mod(12, n^2)^(eulerphi(n))==1, print1(n, ", ")))

a(8)-a(12) from Giovanni Resta, Jan 24 2020

a(13)-a(19) from Giovanni Resta, Jan 27 2020

A241977 Numbers k>1 such that 10^phi(k) == 1 (mod k^2), where phi(n)=A000010(n).

Original entry on oeis.org

3, 487, 1461, 4383, 13149, 39447, 118341, 355023, 56598313, 169794939, 509384817, 1754547703, 5263643109, 7187985751, 15790929327, 21563957253, 27563378431, 33902389487, 47372787981, 50315900257, 64691871759, 82690135293, 101707168461, 150947700771

Offset: 1

Felix Fröhlich, Aug 10 2014

nonn

Giovanni Resta, Table of n, a(n) for n = 1..130 (terms < 10^15)

Cf. A000010, A045616, A077816, A171928, A242958, A242959, A242960, A245529.

Select[Range[400000], PowerMod[10, EulerPhi[#], #^2] == 1 &] (* Amiram Eldar, Oct 16 2023 *)

for(n=2, 1e9, if(Mod(10, n^2)^(eulerphi(n))==1, print1(n, ", ")))

Terms a(12) and beyond from Giovanni Resta, Jan 24 2020

A241978 Numbers n such that 6^phi(n) == 1 (modulo n^2), where phi(n) is Euler's totient function.

Original entry on oeis.org

66161, 330805, 534851, 2674255, 3152573, 10162169, 13371275, 50810845, 54715147, 129255493, 148170931, 254054225, 273575735, 301121113, 383006029, 646277465, 1289402357, 1505605565, 1915030145, 3228193673, 3407931413, 5721301147, 6075008171, 7528027825

Offset: 1

Felix Fröhlich, Aug 10 2014

nonn

a(17) > 10^9.

Giovanni Resta, Table of n, a(n) for n = 1..95 (terms < 10^14)

Cf. A000010, A077816, A212583, A242958, A242959, A242960, A241977, A245529.

Select[Range[65*10^7],PowerMod[6,EulerPhi[#],#^2]==1&] (* Harvey P. Dale, Jan 20 2020 *)

for(n=2, 1e9, if(Mod(6, n^2)^(eulerphi(n))==1, print1(n, ", ")))

Terms a(17) and beyond from Giovanni Resta, Jan 24 2020

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

Original entry on oeis.org

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, ", ")))

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, ", ")))

cp's OEIS Frontend

A242959 Numbers n such that 5^A000010(n) == 1 (mod n^2).

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A242958 Numbers n such that 3^phi(n) == 1 (mod n^2), where phi(n) is Euler's totient function.

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A245529 Numbers n such that 12^phi(n) == 1 (mod n^2), where phi(n) is Euler's totient function.

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A241977 Numbers k>1 such that 10^phi(k) == 1 (mod k^2), where phi(n)=A000010(n).

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A241978 Numbers n such that 6^phi(n) == 1 (modulo n^2), where phi(n) is Euler's totient function.

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

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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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