A257833 -id:A257833 - cp's OEIS frontend

A353937 Smallest b > 1 such that b^(p-1) == 1 (mod p^4) for p = prime(n).

17, 80, 182, 1047, 1963, 239, 4260, 2819, 19214, 2463, 15714, 51344, 20677, 3038, 224444, 189323, 11550, 397575, 201305, 15384, 840838, 1372873, 1576656, 278454, 1721322, 48072, 281007, 119551, 252595, 1001934, 3489507, 2489004, 598987, 3082551, 6136759, 3928984

Offset: 1

Felix Fröhlich, May 12 2022

nonn

Robert Israel, Table of n, a(n) for n = 1..10000

Row k = 4 of A257833.

Cf. similar sequences for p^k: A039678 (k=2), A249275 (k=3), A353938 (k=5), A353939 (k=6), A353940 (k=7), A353941 (k=8), A353942 (k=9), A353943 (k=10).

f:= proc(j) local p,b,i;
  p:= ithprime(j);
  b:= numtheory:-primroot(p^4) &^ (p^3) mod p^4;
  min(seq(b &^i mod p^4, i=1..p-2))
end proc:
f(1):= 17:
map(f, [$1..40]); # Robert Israel, Dec 19 2024

a[n_] := Module[{p = Prime[n], b = 2}, While[PowerMod[b, p - 1, p^4] != 1, b++]; b]; Array[a, 20] (* Amiram Eldar, May 12 2022 *)

a(n) = my(p=prime(n)); for(b=2, oo, if(Mod(b, p^4)^(p-1)==1, return(b)))

from sympy import prime
from sympy.ntheory.residue_ntheory import nthroot_mod
def A353937(n): return 2**4+1 if n == 1 else int(nthroot_mod(1,(p:= prime(n))-1,p**4,True)[1]) # Chai Wah Wu, May 17 2022

A353938 Smallest b > 1 such that b^(p-1) == 1 (mod p^5) for p = prime(n).

Original entry on oeis.org

33, 242, 1068, 1353, 27216, 109193, 15541, 133140, 495081, 1115402, 2754849, 1353359, 649828, 3228564, 2359835, 4694824, 7044514, 28538377, 1111415, 77588426, 16178110, 2553319, 9571390, 158485540, 18664438, 146773512, 45639527, 448251412, 48834112, 141076650

Offset: 1

Felix Fröhlich, May 12 2022

nonn

Row k = 5 of A257833.

Cf. similar sequences for p^k: A039678 (k=2), A249275 (k=3), A353937 (k=4), A353939 (k=6), A353940 (k=7), A353941 (k=8), A353942 (k=9), A353943 (k=10).

a[n_] := Module[{p = Prime[n], b = 2}, While[PowerMod[b, p - 1, p^5] != 1, b++]; b]; Array[a, 12] (* Amiram Eldar, May 12 2022 *)

a(n) = my(p=prime(n)); for(b=2, oo, if(Mod(b, p^5)^(p-1)==1, return(b)))

from sympy import prime
from sympy.ntheory.residue_ntheory import nthroot_mod
def A353938(n): return 2**5+1 if n == 1 else int(nthroot_mod(1,(p:= prime(n))-1,p**5,True)[1]) # Chai Wah Wu, May 17 2022

A353939 Smallest b > 1 such that b^(p-1) == 1 (mod p^6) for p = prime(n).

Original entry on oeis.org

65, 728, 1068, 34967, 284995, 861642, 390112, 333257, 2818778, 42137700, 8078311, 33518159, 92331463, 21583010, 138173066, 8202731, 390421192, 1006953931, 77622331, 270657300, 5915704483, 522911165, 2507851273, 1329885769, 2789067613, 3987072867, 7938255646

Offset: 1

Felix Fröhlich, May 12 2022

nonn

Row k = 6 of A257833.

Cf. similar sequences for p^k: A039678 (k=2), A249275 (k=3), A353937 (k=4), A353938 (k=5), A353940 (k=7), A353941 (k=8), A353942 (k=9), A353943 (k=10).

a[n_] := Module[{p = Prime[n], b = 2}, While[PowerMod[b, p - 1, p^6] != 1, b++]; b]; Array[a, 9] (* Amiram Eldar, May 12 2022 *)

a(n) = my(p=prime(n)); for(b=2, oo, if(Mod(b, p^6)^(p-1)==1, return(b)))

from sympy import prime
from sympy.ntheory.residue_ntheory import nthroot_mod
def A353939(n): return 2**6+1 if n == 1 else int(nthroot_mod(1,(p:= prime(n))-1,p**6,True)[1]) # Chai Wah Wu, May 17 2022

a(25)-a(27) from Jinyuan Wang, May 17 2022

A353940 Smallest b > 1 such that b^(p-1) == 1 (mod p^7) for p = prime(n).

Original entry on oeis.org

129, 2186, 32318, 82681, 758546, 6826318, 21444846, 44702922, 178042767, 393747520, 1548729003, 4741156070, 2203471139, 3242334565, 16609835418, 114175761515, 30338830655, 20115543070, 114457309347, 370162324382, 57877856575, 12692933349, 280646695286, 127762186531

Offset: 1

Felix Fröhlich, May 12 2022

nonn

Row k = 7 of A257833.

Cf. similar sequences for p^k: A039678 (k=2), A249275 (k=3), A353937 (k=4), A353938 (k=5), A353939 (k=6), A353941 (k=8), A353942 (k=9), A353943 (k=10).

a(n) = my(p=prime(n)); for(b=2, oo, if(Mod(b, p^7)^(p-1)==1, return(b)))

from sympy import prime
from sympy.ntheory.residue_ntheory import nthroot_mod
def A353940(n): return 2**7+1 if n == 1 else int(nthroot_mod(1,(p:= prime(n))-1,p**7,True)[1]) # Chai Wah Wu, May 17 2022

a(9)-a(11) from Amiram Eldar, May 12 2022

More terms from Jinyuan Wang, May 17 2022

A353941 Smallest b > 1 such that b^(p-1) == 1 (mod p^8) for p = prime(n).

Original entry on oeis.org

257, 6560, 110443, 2387947, 9236508, 6826318, 112184244, 674273372, 571782680, 8827420195, 46142113101, 85760131222, 287369842623, 120773832179, 83209719751, 1684374218587, 6358345589299, 6305601215112, 5800992744105, 33960226045484, 56924554232879, 11856046381401

Offset: 1

Felix Fröhlich, May 12 2022

nonn

Row k = 8 of A257833.

Cf. similar sequences for p^k: A039678 (k=2), A249275 (k=3), A353937 (k=4), A353938 (k=5), A353939 (k=6), A353940 (k=7), A353942 (k=9), A353943 (k=10).

a(n) = my(p=prime(n)); for(b=2, oo, if(Mod(b, p^8)^(p-1)==1, return(b)))

from sympy import prime
from sympy.ntheory.residue_ntheory import nthroot_mod
def A353941(n): return 2**8+1 if n == 1 else int(nthroot_mod(1,(p:= prime(n))-1,p**8,True)[1]) # Chai Wah Wu, May 17 2022

a(7)-a(8) from Amiram Eldar, May 12 2022

More terms from Jinyuan Wang, May 17 2022

A353942 Smallest b > 1 such that b^(p-1) == 1 (mod p^9) for p = prime(n).

Original entry on oeis.org

513, 19682, 280182, 14906455, 676386984, 822557039, 8185328614, 1835323405, 147534349327, 430099398783, 746688111476, 3054750102760, 9430469115218, 42562034654367, 92084372092298, 28307243117603, 17362132628379, 430275700643181, 478910674129864, 69114209866295

Offset: 1

Felix Fröhlich, May 12 2022

nonn

Row k = 9 of A257833.

Cf. similar sequences for p^k: A039678 (k=2), A249275 (k=3), A353937 (k=4), A353938 (k=5), A353939 (k=6), A353940 (k=7), A353941 (k=8), A353943 (k=10).

a(n) = my(p=prime(n)); for(b=2, oo, if(Mod(b, p^9)^(p-1)==1, return(b)))

from sympy import prime
from sympy.ntheory.residue_ntheory import nthroot_mod
def A353942(n): return 2**9+1 if n == 1 else int(nthroot_mod(1,(p:= prime(n))-1,p**9,True)[1]) # Chai Wah Wu, May 17 2022

a(5)-a(6) from Amiram Eldar, May 12 2022

More terms from Jinyuan Wang, May 17 2022

A353943 Smallest b > 1 such that b^(p-1) == 1 (mod p^10) for p = prime(n).

Original entry on oeis.org

1025, 59048, 3626068, 135967276, 1509748675, 14149342837, 109522148350, 649340249056, 191730243526, 45941644105613, 6359301533362, 24287026146320, 265934493600922, 927939012431924, 1377672497815095, 4440230734662684, 10400007512898615, 12198961352308417

Offset: 1

Felix Fröhlich, May 12 2022

nonn

Row k = 10 of A257833.

Cf. similar sequences for p^k: A039678 (k=2), A249275 (k=3), A353937 (k=4), A353938 (k=5), A353939 (k=6), A353940 (k=7), A353941 (k=8), A353942 (k=9).

a(n) = my(p=prime(n)); for(b=2, oo, if(Mod(b, p^10)^(p-1)==1, return(b)))

from sympy.ntheory.residue_ntheory import nthroot_mod
from sympy import prime
def A353943(n): return 2**10+1 if n == 1 else int(nthroot_mod(1,(p:= prime(n))-1,p**10,True)[1]) # Chai Wah Wu, May 17 2022

a(5)-a(6) from Amiram Eldar, May 12 2022

More terms from Jinyuan Wang, May 17 2022

A258045 Table T(b, m) of largest exponents k such that for p = prime(m) and base b > 1 the congruence b^(p-1) == 1 (mod p^k) is satisfied, or 0 if no such k exists, read by antidiagonals (downwards).

Original entry on oeis.org

0, 1, 1, 1, 0, 0, 1, 1, 1, 2, 1, 1, 1, 1, 0, 1, 2, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 2, 2, 3, 1, 1, 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 0, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 2

Felix Fröhlich, May 26 2015

a(n) > 1 if b appears in row k, column n of the table in A257833 for k > 1 and n > 1.

			T(3, 5) = 2, because the largest Wieferich exponent of prime(5) = 11 in base 3 is 2.
Table starts
b=2:  0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ...
b=3:  1, 0, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1 ...
b=4:  0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ...
b=5:  2, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ...
b=6:  0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ...
b=7:  1, 1, 2, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ...
b=8:  0, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ...
b=9:  3, 0, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1 ...
b=10: 0, 2, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ...
b=11: 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1 ...
b=12: 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ...
b=13: 2, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1 ...
b=14: 0, 1, 1, 0, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1 ...
b=15: 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ...
b=16: 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ...
b=17: 4, 2, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1 ...
b=18: 0, 0, 2, 3, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1 ...
b=19: 1, 2, 1, 3, 1, 2, 1, 0, 1, 1, 1, 1, 1, 2 ...
b=20: 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ...
....
The triangle a(n ,m) begins:
  m 1 2 3 4 5 6 7 8 9 10 11 ...
n
2   0
3   1 1
4   1 0 0
5   1 1 1 2
6   1 1 1 1 0
7   1 2 1 0 0 1
8   1 1 1 1 1 1 0
9   1 1 1 1 1 2 2 3
10  1 1 1 1 1 0 1 0 0
11  1 1 1 1 1 1 1 1 2  1
12  1 1 1 1 1 1 1 1 0  1  0
...

Cf. A001220, A257833.

for(b=2, 20, forprime(p=1, 70, k=0; while(Mod(b, p^k)^(p-1)==1, k++); if(k > 0, k--); print1(k, ", ")); print(""))

a(n, m) = T(m+1, n-m), n >=2, m = 1, 2, ..., n-1. - Wolfdieter Lang, Jun 29 2015

Edited by Wolfdieter Lang, Jun 29 2015

A273433 Smallest base b > 1 such that p = prime(n) satisfies b^(p-1) == 1 (mod p^n).

Original entry on oeis.org

3, 8, 57, 1047, 27216, 861642, 21444846, 674273372, 147534349327, 45941644105613, 244158265828023, 569209871196597077, 15204969799577672558, 2111244131216208612515, 129192323492968413250921, 25814874251189658671192458, 13543096262710710570994132579

Offset: 1

Felix Fröhlich, May 22 2016

nonn

Hiroaki Yamanouchi, Table of n, a(n) for n = 1..46

Cf. A257833.

a(n) = my(p=prime(n), b=2); while(Mod(b, p^n)^(p-1)!=1, b++); b

a(n) = A257833(n, n) for n > 1.

a(9)-a(17) from Hiroaki Yamanouchi, May 26 2016

A281747 Smallest b > 1 such that p = prime(n) satisfies b^(p-1) == 1 (mod p^p).

Original entry on oeis.org

5, 26, 1068, 82681, 5392282366, 11356596271444, 34451905517028761171, 340625514346676110671584, 308318432223607315018221180590, 8566187045843934976180705488213013173127, 1099862052702774330481800364074681495062836757, 8170421001593885871548404108552563632485969048059688187

Offset: 1

Felix Fröhlich, Jan 29 2017

a(n) is the element in row prime(n), column n of the table in A257833.
Is the sequence always nondecreasing, or stronger, is it always increasing?
For odd primes p, if c is a primitive root mod p^p then b == c^(p^(p-1)) (mod p^p) satisfies this.  Thus a(n) < prime(n)^prime(n) for n > 1. - Robert Israel, Jan 30 2017

Robert Israel, Table of n, a(n) for n = 1..76
W. Keller and J. Richstein, Solutions of the congruence a^(p-1) == 1 (mod p^r), Math. Comp. 74 (2005), 927-936.

Cf. A257833.

f:= proc(p) local c,j;
       c:= numtheory:-primroot(p^p);
       min(seq(c &^ (j*p^(p-1)) mod p^p, j=1..p-2))
end proc:
5, seq(f(ithprime(i)),i=2..15); # Robert Israel, Jan 30 2017

Table[b = 2; While[PowerMod[b, (# - 1), #^#] &@ Prime@ n != 1, b++]; b, {n, 4}] (* Michael De Vlieger, Jan 30 2017 *)

a(n) = my(p=prime(n), b=2); while(Mod(b, p^p)^(p-1)!=1, b++); b

More terms from Robert Israel, Jan 30 2017

cp's OEIS Frontend

A353937 Smallest b > 1 such that b^(p-1) == 1 (mod p^4) for p = prime(n).

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A353938 Smallest b > 1 such that b^(p-1) == 1 (mod p^5) for p = prime(n).

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A353939 Smallest b > 1 such that b^(p-1) == 1 (mod p^6) for p = prime(n).

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A353940 Smallest b > 1 such that b^(p-1) == 1 (mod p^7) for p = prime(n).

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A353941 Smallest b > 1 such that b^(p-1) == 1 (mod p^8) for p = prime(n).

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A353942 Smallest b > 1 such that b^(p-1) == 1 (mod p^9) for p = prime(n).

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A353943 Smallest b > 1 such that b^(p-1) == 1 (mod p^10) for p = prime(n).

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A258045 Table T(b, m) of largest exponents k such that for p = prime(m) and base b > 1 the congruence b^(p-1) == 1 (mod p^k) is satisfied, or 0 if no such k exists, read by antidiagonals (downwards).

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A273433 Smallest base b > 1 such that p = prime(n) satisfies b^(p-1) == 1 (mod p^n).

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