A196202 a(n) = 2^(prime(n)-1) mod prime(n)^2.

2, 4, 16, 15, 56, 40, 222, 58, 392, 30, 187, 38, 944, 1076, 2069, 1909, 473, 2197, 671, 143, 4089, 1502, 3985, 535, 5530, 9293, 6078, 1392, 7304, 9380, 2287, 2228, 7262, 4171, 14305, 8457, 12875, 10922, 7850, 520, 8951, 26789, 9551, 20073, 34476, 26866

Offset: 1

Author

Reinhard Zumkeller, Sep 29 2011

Keywords

• nonn
• easy

Comments

a(A049084(A001220(1))) = a(A049084(A001220(2))) = 1.

Examples

			A001220(1)=1093=A000040(183): a(183)=1, or a(A049084(A001220(1)))=1;
A001220(2)=3511=A000040(490): a(490)=1, or a(A049084(A001220(2)))=1.
		

References

• N. G. W. H. Beeger, On a new case of the congruence 2^(p-1) ≡ 1 (p^2), Messenger of Mathematics 51, (1922), p. 149-150
• Paulo Ribenboim, 1093 (Chap 8), in 'My Numbers, My Friends', Springer-Verlag 2000 NY, page 213ff.

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..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]
• Eric Weisstein's World of Mathematics, Wieferich Prime.
• Wikipedia, Wieferich prime.

Crossrefs

Cf. A061286.

Programs

Haskell

import Math.NumberTheory.Moduli (powerMod)
a196202 n = powerMod 2 (p - 1) (p ^ 2) where p = a000040 n
-- Reinhard Zumkeller, May 18 2015

Maple

seq(2 &^ (ithprime(n)-1) mod ithprime(n)^2, n=1..1000); # Robert Israel, Aug 03 2014

Mathematica

PowerMod[2,#-1,#^2]&/@Prime[Range[50]] (* Harvey P. Dale, Apr 25 2012 *)

PARI

forprime(p=2, 1e2, print1(lift(Mod(2, p^2)^(p-1)), ", ")) \\ Felix Fröhlich, Aug 03 2014