A245529 -id:A245529 - cp's OEIS frontend

A111027 Wieferich primes in base 12.

2693, 123653

Offset: 1

Walter Kehowski, Oct 05 2005

I have searched up to the 9 millionth prime, 160481183 and gave up trying to find a third term. The sequence is conjectured to be infinite. If the behavior is similar to base 10, A045616, then the next term could be greater than 2*10^11. In base 12 with X for ten and E for eleven the sequence is [1685, 5E685] so it would be interesting to see if the third term ends in 685 as well. These primes are also the Wieferich numbers in base 12: 12^phi(n) = 1 mod n^2.

Richard Fischer has carried this search to 4.8 * 10^13 (as of January 2014). - John Blythe Dobson, Mar 06 2014

Amir Akbary and Sahar Siavashi, The Largest Known Wieferich Numbers, INTEGERS, 18(2018), A3. See Table 1 p. 5.
Richard Fischer, Fermatquotient B^(P-1) == 1 (mod P^2).

Cf. A001220, A039951, A045616, A077815, A077816, A245529.

WP:=[]: for z from 1 to 1 do for k from 1 to 9000000 do p:=ithprime(k); if 12 &^(p-1) mod p^2 = 1 then WP:=[op(WP),p]; printf("p=%d, ",p); fi; if k mod 10^5 = 0 then printf("k=%d, ",k); fi; od; od; WP;

Select[Prime[Range[1000000]], PowerMod[12, # - 1, #^2] == 1 &] (* Robert Price, May 17 2019 *)

12^(p-1) == 1 mod p^2

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, ", ")))

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A111027 Wieferich primes in base 12.

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

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