A242742 -id:A242742 - cp's OEIS frontend

A249275 a(n) is the smallest b > 1 such that p = prime(n) satisfies b^(p-1) == 1 (mod p^3).

9, 26, 57, 18, 124, 239, 158, 333, 42, 1215, 513, 691, 1172, 3038, 295, 1468, 2511, 15458, 3859, 6372, 923, 1523, 5436, 1148, 412, 4943, 4432, 5573, 476, 68, 21304, 30422, 6021, 8881, 33731, 25667, 3868, 3170, 17987, 26626, 43588, 7296, 14628, 22076, 138057

Offset: 1

Felix Fröhlich, Oct 24 2014

nonn

a(n) >= A039678(n) for all n.

Vincenzo Librandi, Table of n, a(n) for n = 1..100

Cf. A039678, A242742.

Array[Block[{b = 2}, While[PowerMod[b, # - 1, #^3] != 1, b++]; b] &@ Prime@ # &, 45] (* Michael De Vlieger, Nov 25 2018 *)
dpa[n_]:=Module[{p=Prime[n], a=9}, While[PowerMod[a, p - 1, p^3]!=1, a++]; a]; Array[dpa, 50] (* Vincenzo Librandi, Nov 30 2018 *)

a(n) = my(p=prime(n)); for(b=2, oo, if(Mod(b, p^3)^(p-1)==1, return(b)))

from sympy import prime
def a(n):
    b, p = 2, prime(n)
    p3 = p**3
    while pow(b, p-1, p3) != 1: b += 1
    return b
print([a(n) for n in range(1, 46)]) # Michael S. Branicky, Sep 26 2021

from sympy import prime
from sympy.ntheory.residue_ntheory import nthroot_mod
def A249275(n): return 2**3+1 if n == 1 else int(nthroot_mod(1,(p:= prime(n))-1,p**3,True)[1]) # Chai Wah Wu, May 17 2022

Edited by Felix Fröhlich, Nov 24 2018

A256517 Let c be the n-th composite number. Then a(n) is the smallest base b > 1 such that b^(c-1) == 1 (mod c^2), i.e., such that c is a 'Wieferich pseudoprime'.

Original entry on oeis.org

17, 37, 65, 80, 101, 145, 197, 26, 257, 325, 401, 197, 485, 577, 182, 677, 728, 177, 901, 1025, 485, 1157, 99, 1297, 1445, 170, 1601, 1765, 1937, 82, 2117, 2305, 1047, 2501, 577, 529, 2917, 1451, 3137, 721, 3365, 3601, 3845, 244, 4097, 99, 1945, 4625, 530

Offset: 1

Felix Fröhlich, Apr 01 2015

nonn

Chai Wah Wu, Table of n, a(n) for n = 1..10000 (terms 1..3000 from Felix Fröhlich)

Cf. A039678, A242742, A244752, A255885.

Cf. A185103, A002808.

c = Select[Range@ 69, CompositeQ]; f[c_] := Block[{b = 2}, While[Mod[b^(c - 1), c^2] != 1, b++]; b]; f /@ c (* Michael De Vlieger, Apr 03 2015 *)

forcomposite(c=1, 1e3, b=2; while(Mod(b, c^2)^(c-1)!=1, b++); print1(b, ", "))

from sympy import composite
from sympy.ntheory.residue_ntheory import nthroot_mod
def A256517(n):
    z = nthroot_mod(1,(c := composite(n))-1,c**2,True)
    return int(z[0]+c**2 if len(z) == 1 else z[1]) # Chai Wah Wu, May 18 2022

a(n) = A185103(A002808(n)-1). - Bill McEachen, Nov 27 2021

A323602 Smallest b > 1 not already in the sequence such that b^(c-1) == 1 (mod c), i.e., c is a base-b Fermat pseudoprime, where c is the n-th composite number (A002808).

Original entry on oeis.org

5, 7, 9, 8, 11, 13, 15, 4, 17, 19, 21, 20, 23, 25, 18, 27, 26, 29, 31, 33, 10, 35, 6, 37, 39, 14, 41, 43, 45, 19, 47, 49, 30, 51, 16, 53, 55, 34, 57, 56, 59, 61, 63, 62, 65, 12, 67, 69, 22, 71, 73, 75, 74, 77, 76, 79, 81, 80, 83, 85, 38, 87, 28, 89, 91, 3, 93

Offset: 1

Felix Fröhlich, Jan 19 2019

nonn

Is this a permutation of the positive integers > 1?

Cf. A002808, A242742, A259234, A323603.

my(v=vector(1)); forcomposite(c=1, 50, my(b=2); while(Mod(b, c)^(c-1)!=1, b++; if(Mod(b, c)^(c-1)==1, for(k=1, #v, if(b==v[k], b++)))); v=concat(v, b); print1(v[#v], ", "))

A309383 a(n) is the smallest b > 1 such that when c is equal to any of the first n composites the congruence b^(c-1) == 1 (mod c) is satisfied, i.e., smallest b larger than 1 such that any member of the set of smallest n composites is a base-b Fermat pseudoprime.

Original entry on oeis.org

5, 13, 25, 73, 361, 361, 2521, 2521, 5041, 5041, 5041, 5041, 55441, 55441, 277201, 3603601, 10810801, 10810801, 10810801, 21621601, 21621601, 367567201, 367567201, 367567201

Offset: 1

Felix Fröhlich, Jul 27 2019

			For n = 4: The four smallest composites are 4, 6, 8, 9 and for those four values of c the congruence b^(c-1) == 1 (mod c) is satisfied with b = 73. Since 73 is the smallest such value of b > 1, a(4) = 73.

Cf. A242742, A256236.

composites(n) = my(v=[]); forcomposite(c=1, , v=concat(v, [c]); if(#v >= n, return(v)))
a(n) = my(cp=composites(n)); for(b=2, oo, for(k=1, #cp, if(Mod(b, cp[k])^(cp[k]-1)!=1, break, if(k==#cp, return(b)))))

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A249275 a(n) is the smallest b > 1 such that p = prime(n) satisfies b^(p-1) == 1 (mod p^3).

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A256517 Let c be the n-th composite number. Then a(n) is the smallest base b > 1 such that b^(c-1) == 1 (mod c^2), i.e., such that c is a 'Wieferich pseudoprime'.

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A323602 Smallest b > 1 not already in the sequence such that b^(c-1) == 1 (mod c), i.e., c is a base-b Fermat pseudoprime, where c is the n-th composite number (A002808).

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A309383 a(n) is the smallest b > 1 such that when c is equal to any of the first n composites the congruence b^(c-1) == 1 (mod c) is satisfied, i.e., smallest b larger than 1 such that any member of the set of smallest n composites is a base-b Fermat pseudoprime.

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