A241978 -id:A241978 - 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, ", ")))

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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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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