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
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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 *)
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a(n) = my(p=prime(n)); for(b=2, oo, if(Mod(b, p^5)^(p-1)==1, return(b)))
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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
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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 *)
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a(n) = my(p=prime(n)); for(b=2, oo, if(Mod(b, p^6)^(p-1)==1, return(b)))
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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
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
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a(n) = my(p=prime(n)); for(b=2, oo, if(Mod(b, p^7)^(p-1)==1, return(b)))
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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
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
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a(n) = my(p=prime(n)); for(b=2, oo, if(Mod(b, p^8)^(p-1)==1, return(b)))
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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
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
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a(n) = my(p=prime(n)); for(b=2, oo, if(Mod(b, p^9)^(p-1)==1, return(b)))
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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
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
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a(n) = my(p=prime(n)); for(b=2, oo, if(Mod(b, p^10)^(p-1)==1, return(b)))
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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
A286816
Smallest b such that the k consecutive primes starting with prime(n) are all base-b Wieferich primes, i.e., satisfy b^(p-1) == 1 (mod p^2). Square array A(n, k), read by antidiagonals downwards.
Original entry on oeis.org
5, 17, 8, 449, 26, 7, 557, 226, 18, 18, 19601, 1207, 1207, 148, 3, 132857, 54568, 1451, 606, 239, 19, 4486949, 2006776, 13543, 13543, 3469, 249, 38, 126664001, 20950343, 296449, 296449, 24675, 653, 423, 28, 2363321449, 230695118, 23250274, 17134811, 3414284, 39016, 5649, 28, 28, 5229752849, 5229752849, 882345432, 741652533, 36763941, 14380864, 217682, 26645, 63, 14
Offset: 1
The sequence of base-226 Wieferich primes starts 3, 5, 7, 97, 157, ... Since 226 is the smallest b such that the three consecutive primes starting with prime(2) = 3 are base-b Wieferich primes, A(2, 3) = 226.
Array starts:
n=1: 5, 17, 449, 557, 19601, 132857
n=2: 8, 26, 226, 1207, 54568, 2006776
n=3: 7, 18, 1207, 1451, 13543, 296449
n=4: 18, 148, 606, 13543, 296449, 17134811
n=5: 3, 239, 3469, 24675, 3414284, 36763941
n=6: 19, 249, 653, 39016, 14380864, 34998229
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primevec(initialp, vecsize) = my(v=[initialp]); while(#v < vecsize, v=concat(v, nextprime(v[#v]+1))); v
a(n, k) = my(v=primevec(prime(n), k), b=2, i=0); while(1, for(x=1, #v, if(Mod(b, v[x]^2)^(v[x]-1)!=1, i++; break)); if(i==0, return(b)); b++; i=0)
array(rows, cols) = for(s=1, rows, for(t=1, cols, print1(a(s, t), ", ")); print(""))
array(5, 6) \\ print 5 X 6 array
A344828
a(n) is the smallest b > 1 such that prime(n), prime(n+1), prime(n+2) and prime(n+3) are all base-b Wieferich primes.
Original entry on oeis.org
557, 1207, 1451, 13543, 24675, 39016, 217682, 165407, 1357748, 399254, 1146590, 325346, 1895206, 3365181, 4674177, 21251205, 40698745, 6795147, 36463448, 12717474, 54383927, 7411274, 35989426, 101112784, 86045167, 13128506, 276293632, 169093089, 223680564, 137073637
Offset: 1
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a(n) = my(v=[prime(n)]); while(#v < 4, v=concat(v, nextprime(v[#v]+1))); for(b=2, oo, for(k=1, #v, if(Mod(b, v[k]^2)^(v[k]-1)!=1, break, if(k==#v, return(b)))))
A344829
a(n) is the smallest b > 1 such that prime(n), prime(n+1), prime(n+2), prime(n+3) and prime(n+4) are all base-b Wieferich primes.
Original entry on oeis.org
19601, 54568, 13543, 296449, 3414284, 14380864, 3727271, 7916603, 65097619, 13793462, 152541840, 30495845, 91779237, 183068599, 558175167, 40698745, 825287029, 2151529020, 6271678163, 1266687934, 3149182509, 989067909, 10785363668, 18739432977, 4877709531, 24531035970, 11683733786, 52383593584
Offset: 1
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a(n) = my(v=[prime(n)]); while(#v < 5, v=concat(v, nextprime(v[#v]+1))); for(b=2, oo, for(k=1, #v, if(Mod(b, v[k]^2)^(v[k]-1)!=1, break, if(k==#v, return(b)))))
A344830
a(n) is the smallest b > 1 such that prime(n), prime(n+1), prime(n+2), prime(n+3), prime(n+4) and prime(n+5) are all base-b Wieferich primes.
Original entry on oeis.org
132857, 2006776, 296449, 17134811, 36763941, 34998229, 31239565, 968576295, 1027038511, 287811239, 15022368222, 33452659960, 19477999997, 132892045949, 47341045210, 32176849766, 106967760951, 303459122992, 20216391690, 1411108384416, 219517083156, 361244580521, 455588981749
Offset: 1
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a(n) = my(v=[prime(n)]); while(#v < 6, v=concat(v, nextprime(v[#v]+1))); for(b=2, oo, for(k=1, #v, if(Mod(b, v[k]^2)^(v[k]-1)!=1, break, if(k==#v, return(b)))))