1,1

a(4k+1) = 2. a(n) = A096082(n) for all n that are not of the form 4k+1. a(35)-a(46) = {3,66161,2,17,8039,11,2,23,5,3,2,3}. a(48)-a(65) = {7,2,7,5,461,2,19,3,647,2,131,2777,29,2,3,23,3,2}. a(67)-a(71) = {7,5,2,13,3}. a(73)-a(87) = {2,5,17,5,2,43,7,3,2,3,4871,163,2,68239,1999}. a(89) = 2. a(91)-a(122) = {3,727,2,11,2137,109,2,3,5,3,2,7559,24490789,313,2,79399,3,3761,2,17,131,11,2,9181,31,3,2,3,1741,11,2,11}. a(34), a(47), a(66), a(72), a(88), a(90) are currently unknown. - Alexander Adamchuk (alex(AT)kolmogorov.com), Nov 27 2006

Note that a(n) = 1093 for n = {2,4,16,256}, a(n) = 66161 for n = {6,36,216}, a(n) = 2693 for n = {12,144}, a(n) = 29 for n = {14,196}, a(n) = 281 for n = {20,400}. It appears that there is a pattern such that in many cases if a(n) = p then a(n^2) = p too. In some cases as for n = 6 a(n^3) = p too. Update: a(n) is currently unknown for n = {47,72,139,160,186,187,200,203,222,231,304,311,335,347,355,387,435,454,464,542,546,552,554,594,610,639,648,662,666,715,758,560,772,798,804,808,612,858,860,871,886,983,986,...}. - Alexander Adamchuk (alex(AT)kolmogorov.com), Nov 30 2006

C. K. Caldwell, The Prime Glossary, Fermat quotient.

W. Keller and J. Richstein, Fermat quotients q_p(a) that are divisible by p.

Helmut Richter, Table of all known a(n) up to n = 1000 http://www.lrz-muenchen.de/~hr/tmp/A039951.txt

Cf. A001220, A045616.

Cf. A096082 = Smallest ODD prime p such that p^2 | n^(p-1) - 1, or 0 if no such p exists. Cf. A001220 = Wieferich primes p: p^2 divides 2^(p-1) - 1. Cf. A014127 = Primes p such that p^2 divides 3^(p-1) - 1. Cf. A123692 = Primes p such that p^2 divides 5^(p-1) - 1. Cf. A123693 = Primes p such that p^2 divides 7^(p-1) - 1.

Sequence in context: A111203 A108963 A013543 this_sequence A135618 A119554 A036104

Adjacent sequences: A039948 A039949 A039950 this_sequence A039952 A039953 A039954

nonn

David W. Wilson (davidwwilson(AT)comcast.net)

a(34)-a(46) from Helmut Richter (richter(AT)lrz.de), May 17 2004. a(47) > 10^9.

Entry revised by njas, Nov 30 2006