A002323 ((2^m - 1) / p) mod p, where p = prime(n) and m = ord(2,p). (Formerly M2223 N0882)

1, 3, 1, 5, 3, 15, 3, 20, 1, 1, 1, 32, 37, 22, 36, 8, 36, 10, 1, 7, 49, 48, 23, 77, 92, 81, 13, 95, 49, 1, 17, 95, 30, 96, 66, 132, 67, 107, 3, 50, 148, 25, 52, 175, 167, 109, 143, 201, 99, 30, 13, 207, 200, 255, 64, 260, 190, 208, 159, 208, 78, 98, 243, 60

Offset

2,2

Comments

a(n) = 0 if and only if prime(n) is a Wieferich prime (A001220). - Eric M. Schmidt, Feb 23 2015

References

D. H. Lehmer, Guide to Tables in the Theory of Numbers. Bulletin No. 105, National Research Council, Washington, DC, 1941, pp. 7-10.

W. Meißner, Über die Teilbarkeit von 2^p-2 durch das Quadrat der Primzahl p=1093, Sitzungsberichte der Königlich Preußischen Akadamie der Wissenschaften, Berlin, 35 (1913), 663-667.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Eric M. Schmidt, Table of n, a(n) for n = 2..10000

W. Meissner, Über die Teilbarkeit von 2^p-2 durch das Quadrat der Primzahl p = 1093, Sitzungsberichte Königlich Preussischen Akadamie Wissenschaften Berlin, 35 (1913), 663-667. [Annotated scanned copy]

Example

For p = prime(3) = 5, we find that m = 4 is the smallest positive integer for which 2^m - 1 is divisible by p. So a(3) = ((2^4 - 1) / 5) mod 5 = 3. - Eric M. Schmidt, Jun 21 2013

Mathematica

Table[p = Prime[n]; Mod[(2^MultiplicativeOrder[2, p] - 1)/p, p], {n, 2, 100}] (* T. D. Noe, Jun 21 2013 *)

Prog

(Sage) def A002323(n) : p = nth_prime(n); return (2^(Mod(2, p).multiplicative_order()) - 1) // p % p # Eric M. Schmidt, Jun 21 2013
(PARI) a(n) = my(p=prime(n)); (lift(Mod(2, p^2)^znorder(Mod(2, p)))-1)/p \\ Jeppe Stig Nielsen, May 30 2023

Crossrefs

Cf. A001220, A001917.

Sequence in context: A289891 A289094 A171382 * A294640 A365671 A200920

Adjacent sequences: A002320 A002321 A002322 * A002324 A002325 A002326

Keyword

nonn,easy

Author

N. J. A. Sloane

Extensions

Proper definition added by and more terms from Eric M. Schmidt, Jun 21 2013

Status

approved