%I #37 Oct 24 2024 02:20:22

%S 2,4,16,15,56,40,222,58,392,30,187,38,944,1076,2069,1909,473,2197,671,

%T 143,4089,1502,3985,535,5530,9293,6078,1392,7304,9380,2287,2228,7262,

%U 4171,14305,8457,12875,10922,7850,520,8951,26789,9551,20073,34476,26866

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

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

%D 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

%D Paulo Ribenboim, 1093 (Chap 8), in 'My Numbers, My Friends', Springer-Verlag 2000 NY, page 213ff.

%H Reinhard Zumkeller, <a href="/A196202/b196202.txt">Table of n, a(n) for n = 1..10000</a>

%H W. Meissner, <a href="/A001917/a001917.pdf">Über die Teilbarkeit von 2^p-2 durch das Quadrat der Primzahl p = 1093</a>, Sitzungsberichte Königlich Preussischen Akadamie Wissenschaften Berlin, 35 (1913), 663-667. [Annotated scanned copy]

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/WieferichPrime.html">Wieferich Prime</a>.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Wieferich_prime">Wieferich prime</a>.

%e A001220(1)=1093=A000040(183): a(183)=1, or a(A049084(A001220(1)))=1;

%e A001220(2)=3511=A000040(490): a(490)=1, or a(A049084(A001220(2)))=1.

%p seq(2 &^ (ithprime(n)-1) mod ithprime(n)^2, n=1..1000); # _Robert Israel_, Aug 03 2014

%t PowerMod[2,#-1,#^2]&/@Prime[Range[50]] (* _Harvey P. Dale_, Apr 25 2012 *)

%o (PARI) forprime(p=2, 1e2, print1(lift(Mod(2, p^2)^(p-1)), ", ")) \\ _Felix Fröhlich_, Aug 03 2014

%o (Haskell)

%o import Math.NumberTheory.Moduli (powerMod)

%o a196202 n = powerMod 2 (p - 1) (p ^ 2) where p = a000040 n

%o -- _Reinhard Zumkeller_, May 18 2015

%Y Cf. A061286.

%K nonn,easy

%O 1,1

%A _Reinhard Zumkeller_, Sep 29 2011