A256236 -id:A256236 - cp's OEIS frontend

A339533 Numbers b > 1 such that the smallest four primes, i.e., 2, 3, 5 and 7 are base-b Wieferich primes.

557, 901, 1549, 2449, 4049, 5293, 5849, 6193, 7057, 7957, 8801, 9701, 12349, 13249, 14093, 14993, 15857, 16201, 16757, 18001, 19601, 20501, 21149, 21493, 22049, 23257, 23293, 24893, 25057, 25793, 26657, 27557, 28549, 30349, 31949, 32849, 33301, 34201, 35801

Offset: 1

Felix Fröhlich, Dec 08 2020

nonn

Cf. A256236. Row 1 of A319061.

Cf. smallest k primes are base-b Wieferich primes: A339531 (k=2), A339532 (k=3), A339534 (k=5), A339535 (k=6), A339536 (k=7), A339537 (k=8).

Select[Range[2, 36000], Function[b, AllTrue[{2, 3, 5, 7}, PowerMod[b, (# - 1), #^2] == 1 &]]] (* Michael De Vlieger, Dec 10 2020 *)

is(n) = forprime(p=1, 7, if(Mod(n, p^2)^(p-1)!=1, return(0))); 1

A339534 Numbers b > 1 such that the smallest five primes, i.e., 2, 3, 5, 7 and 11 are base-b Wieferich primes.

Original entry on oeis.org

19601, 22049, 48149, 52057, 67357, 84457, 85193, 93493, 104057, 113257, 115757, 132857, 160849, 171793, 186101, 198449, 206749, 209249, 224549, 228457, 235493, 261593, 265501, 280801, 297901, 317501, 326701, 329201, 329849, 346301, 359857, 374293, 374849

Offset: 1

Felix Fröhlich, Dec 08 2020

nonn

Cf. A256236. Row 1 of A319062.

Cf. smallest k primes are base-b Wieferich primes: A339531 (k=2), A339532 (k=3), A339533 (k=4), A339535 (k=6), A339536 (k=7), A339537 (k=8).

Select[Range[2, 375000], Function[b, AllTrue[{2, 3, 5, 7, 11}, PowerMod[b, (# - 1), #^2] == 1 &]]] (* Michael De Vlieger, Dec 10 2020 *)

is(n) = forprime(p=1, 11, if(Mod(n, p^2)^(p-1)!=1, return(0))); 1

A339535 Numbers b > 1 such that the smallest six primes, i.e., 2, 3, 5, 7, 11 and 13 are base-b Wieferich primes.

Original entry on oeis.org

132857, 171793, 261593, 618301, 700993, 997757, 1211201, 1365857, 1388593, 1542293, 1557593, 1681649, 1692557, 1906001, 2086793, 2124757, 2245357, 2293757, 2341349, 2443501, 2822957, 3025457, 3036401, 3435193, 3554657, 3569257, 3814649, 4028093, 4048901

Offset: 1

Felix Fröhlich, Dec 08 2020

nonn

Cf. A256236. Row 1 of A319063.

Cf. smallest k primes are base-b Wieferich primes: A339531 (k=2), A339532 (k=3), A339533 (k=4), A339534 (k=5), A339536 (k=7), A339537 (k=8).

Select[Range[2, 4100000], Function[b, AllTrue[{2, 3, 5, 7, 11, 13}, PowerMod[b, (# - 1), #^2] == 1 &]]] (* Michael De Vlieger, Dec 10 2020 *)

is(n) = forprime(p=1, 13, if(Mod(n, p^2)^(p-1)!=1, return(0))); 1

A339536 Numbers b > 1 such that the smallest seven primes, i.e., 2, 3, 5, 7, 11, 13 and 17 are base-b Wieferich primes.

Original entry on oeis.org

4486949, 4651993, 4941649, 5571593, 11903257, 19397501, 19841257, 19942001, 20610901, 21530501, 25793893, 42969601, 44404093, 46336949, 51509701, 52786493, 53740457, 54561401, 56120257, 56904857, 63789749, 64028593, 65497301, 65666449, 68441957, 72582101

Offset: 1

Felix Fröhlich, Dec 08 2020

nonn

Cf. A256236. Row 1 of A319064.

Cf. smallest k primes are base-b Wieferich primes: A339531 (k=2), A339532 (k=3), A339533 (k=4), A339534 (k=5), A339535 (k=6), A339537 (k=8).

is(n) = forprime(p=1, 17, if(Mod(n, p^2)^(p-1)!=1, return(0))); 1

A339537 Numbers b > 1 such that the smallest eight primes, i.e., 2, 3, 5, 7, 11, 13, 17 and 19 are base-b Wieferich primes.

Original entry on oeis.org

126664001, 133487693, 141313157, 236176001, 242883757, 356977349, 358254649, 383691493

Offset: 1

Felix Fröhlich, Dec 08 2020

Cf. A256236. Row 1 of A319065.

Cf. smallest k primes are base-b Wieferich primes: A339531 (k=2), A339532 (k=3), A339533 (k=4), A339534 (k=5), A339535 (k=6), A339536 (k=7).

is(n) = forprime(p=1, 19, if(Mod(n, p^2)^(p-1)!=1, return(0))); 1

A258787 Triangle read by rows: T(n, k) = smallest base b > 1 such that p = prime(n) is the k-th base-b Wieferich prime for k = 1, 2, 3, ..., n.

Original entry on oeis.org

5, 8, 17, 7, 26, 449, 30, 18, 197, 557, 3, 9, 118, 1207, 19601, 22, 146, 19, 361, 8201, 132857, 38, 40, 224, 249, 4625, 296449, 4486949, 54, 28, 68, 99, 4033, 4625, 296449, 126664001, 42, 130, 28, 118, 557, 8201, 997757, 24800401, 2363321449, 14, 41, 374, 1745, 901, 46826, 217682, 9312157, 758427193, 5229752849

Offset: 1

Felix Fröhlich, Jun 10 2015

			T(4, 3) = 197, because 197 is the smallest base b such that p = prime(4) = 7 is the 3rd base-b Wieferich prime.
Triangle T(n, k) starts:
  5;
  8,  17;
  7,  26,  449;
  30, 18,  197, 557;
  3,  9,   118, 1207, 19601;
  22, 146, 19,  361,  8201,  132857;
  38, 40,  224, 249,  4625,  296449, 4486949;
  54, 28,  68,  99,   4033,  4625,   296449,  126664001;
  42, 130, 28,  118,  557,   8201,   997757,  24800401,  2363321449;

Cf. A256236 (diagonal). A286816.

nextwiefbase(n, a) = a++; while(Mod(a, n^2)^(n-1)!=1, a++); a
wiefrank(n, a) = i=0; forprime(p=1, n, if(Mod(a, p^2)^(p-1)==1, i++)); i
trianglerows(n) = i=1; while(i <= n, p=prime(i); for(k=1, i, b=2; while(wiefrank(p, b)!=k, b=nextwiefbase(p, b)); print1(b, ", ")); print(""); i++)
trianglerows(9) \\ print first nine rows of the triangle

More terms from Max Alekseyev, Oct 14 2023

A270776 Smallest non-Wieferich prime to base n, i.e., smallest prime p such that n^(p-1) != 1 (mod p^2).

Original entry on oeis.org

2, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2, 5, 2, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2, 5, 2, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2, 5, 2, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2, 5, 2, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2

Offset: 2

Felix Fröhlich, Mar 22 2016

nonn

A256236 gives the smallest i such that a(i) = A000040(n).
a(n) > 2 iff A039951(n) = 2.
a(n) > 3 iff A268352(n) = 3.
Does every prime appear in the sequence?
It is easy to see that the answer to the previous question is "yes" if and only if A256236 is infinite.
The ABC-(k, Epsilon) conjecture with k >= 2 and Epsilon > 0 such that 1/(1/Epsilon + 1) + 1/k  <= log(2)/(24*log(a)) implies that a(n) exists for all n (cf. Broughan, 2006; theorem 5.6).

			The sequence of base-17 Wieferich primes (A128668) starts 2, 3, 46021. Thus the smallest non-Wieferich prime to base 17 is 5 and hence a(17) = 5.

Felix Fröhlich, Table of n, a(n) for n = 2..10000
K. A. Broughan, Relaxations of the abc conjecture using integer k'th roots, New Zealand Journal of Mathematics, 35 (2006), 121-136.

Cf. A039951, A256236, A268352.

A270776[n_] := NestWhile[#+1 &, 2, CompositeQ[#] || PowerMod[n, #-1, #^2] == 1 &];
Array[A270776, 100, 2] (* Paolo Xausa, Aug 15 2025 *)

a(n) = forprime(p=1, , if(Mod(n, p^2)^(p-1)!=1, return(p)))

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)))))

cp's OEIS Frontend

A339533 Numbers b > 1 such that the smallest four primes, i.e., 2, 3, 5 and 7 are base-b Wieferich primes.

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A339534 Numbers b > 1 such that the smallest five primes, i.e., 2, 3, 5, 7 and 11 are base-b Wieferich primes.

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A339535 Numbers b > 1 such that the smallest six primes, i.e., 2, 3, 5, 7, 11 and 13 are base-b Wieferich primes.

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A339536 Numbers b > 1 such that the smallest seven primes, i.e., 2, 3, 5, 7, 11, 13 and 17 are base-b Wieferich primes.

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A339537 Numbers b > 1 such that the smallest eight primes, i.e., 2, 3, 5, 7, 11, 13, 17 and 19 are base-b Wieferich primes.

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A258787 Triangle read by rows: T(n, k) = smallest base b > 1 such that p = prime(n) is the k-th base-b Wieferich prime for k = 1, 2, 3, ..., n.

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A270776 Smallest non-Wieferich prime to base n, i.e., smallest prime p such that n^(p-1) != 1 (mod p^2).

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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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