cp's OEIS Frontend

Prime numbers p such that p^2 divides 22^(p-1) - 1.

Next term, if it exists, is larger than 8.72*10^13.

492366587 was found by Montgomery (cf. Montgomery, 1993). - Felix Fröhlich, Jan 30 2018

Prime numbers p such that p^2 divides 26^(p-1) - 1.

No more terms up to 9.8*10^13.

Prime numbers p such that p^2 divides 30^(p-1) - 1.

No more terms up to 9.8*10^13.

There are two definitions of the periodic part: zeros may either begin or end the periodic part. For example, for 1/11 = 0.0909090..., the periodic part could be either 09 or 90. This sequence assumes that the zeros are at the beginning of the periodic part. See A179267 for the case of zeros at the end of the periodic part. The prime numbers in this sequence are in A045616. The three numbers following 487 are 3*487, 9*487, and 27*487. There are no other multiples of 487 here because 3 and 487 are the only prime factors of 10^486-1 that occur to a power greater than 1. - T. D. Noe, Jul 06 2010

a(n) = Smallest prime such that n appears in A143548. - Eric Chen, Nov 26 2014

The square of a(n) is the smallest squared prime that is a pseudoprime (> n) in base n; for example, a(2) = 1093, and 1093^2 = 1194649 is the smallest squared prime that is pseudoprime in base 2. - Eric Chen, Nov 26 2014

Is a(n) defined for all n? - Eric Chen, Nov 26 2014

Does every prime appear in this sequence? - Eric Chen, Nov 26 2014

a(22)..a(28) = {13, 13, 5, 20771, 71, 11, 19}, a(30)..a(46) = {7, 7, 1093, 233, 46145917691, 1613, 66161, 77867, 17, 8039, 11, 29, 23, 103, 229, 1283, 829}, a(48)..a(49) = {7, 491531}, a(51)..a(60) = {41, 461, 47, 19, 30109, 647, 47699, 131, 2777, 29}, a(62)..a(71) = {19, 23, 1093, 17, 89351671, 47, 19, 19, 13, 47}, a(74)..a(81) = {1251922253819, 17, 37, 32687, 43, 263, 13, 11}, a(83)..a(100) = {4871, 163, 11779, 68239, 1999, 2535619637, 13, 6590291053, 293, 727, 509, 11, 2137, 109, 2914393, 28627, 13, 487}; a(n) is currently unknown for n = {21, 29, 47, 50, 61, 72, 73, 82, 126, 132, 154, 186, 187, 188, 200, 203, 222, 231, 237, 301, 304, 309, 311, 327, 335, 347, 351, 355, 357, 435, 441, 454, 458, 496, 541, 542, 546, 554, 570, 593, 609, 610, 639, 640, 654, 662, 668, 674, 692, 697, 698, 701, 718, 724, 725, 727, 733, 743, 760, 772, 775, 777, 784, 798, 807, 808, 812, 829, 841, 858, 860, 871, 883, 912, 919, 944, 980, 983, 986, ...}. - Eric Chen, Nov 26 2014

a(21) > 3.4 * 10^13. - Eric Chen, Nov 26 2014

Numbers k that are not squarefree and satisfy 10^(k-1) == 1 (mod k).

Any term is divisible by the square of a base-10 Wieferich prime (A045616 = {3, 487, 56598313, ...}).

Intersection of A005939 and A013929.

There are two definitions of the periodic part: zeros may either begin or end the periodic part. For example, for 1/11 = 0.0909090..., the periodic part could be either 09 or 90. This sequences assumes that the zeros are at the end of the periodic part. See A171928 for the case of zeros at the beginning of the periodic part.

Note that the terms following 487 are divisible by 487. This pattern continues until the prime number 56598313, which is another member of A045616.

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)?