The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation. Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 56th year, we are closing in on 350,000 sequences, and we’ve crossed 9,700 citations (which often say “discovered thanks to the OEIS”). Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A261117 Smallest positive integer b such that b^(2^n)+1 is divisible by the square of A035089(n+1). 2

%I

%S 8,7,110,40,1497,894,315,48,166107,95853,63609,71589,492348,209628,

%T 388440,48853,6118793,2684186,25787045,49643800,54302036,3969770538,

%U 17592956651,7347360617,991255542,8249087392,11518171450,51385581002,2268777293,21252616802,2822082710511

%N Smallest positive integer b such that b^(2^n)+1 is divisible by the square of A035089(n+1).

%C For given n, if A035089(n+1) exists (which is true by Dirichlet's theorem on arithmetic progressions), then a(n) exists. Proof: p := A035089(n+1) is a prime of the form p=k*2^(n+1)+1, then the group (Z/(p^2)Z)* is cyclic of order p*(p-1) = p*k*2^(n+1). It therefore has an element b of order exactly 2^(n+1). For that b we have then b^(2^n) == -1 (mod p^2).

%C For given n, a(n) is not necessarily the smallest b such that b^(2^n)+1 is nonsquarefree; see A260824.

%e Consider n=4, hence generalized Fermat numbers b^16+1. The first prime (A035089(4+1)) of the form 32*k+1 is 97. It follows that 97 is the smallest prime whose square divides a number of the form b^16+1. The first time 97^2 divides b^16+1 is for b=1497. Hence a(4)=1497. However, A260824(4) is smaller, A260824(4)=392. This is because already 392^16+1 is nonsquarefree (but the prime with a square dividing it, 769, exceeds 97).

%o (PARI) a(n)=for(k=1,10^10,p=(k<<(n+1))+1;if(isprime(p),break()));for(b=1,p^2,b%p!=0&Mod(b,p^2)^(1<<n)==-1&return(b))

%o (PARI) a(n)=for(k=1, 10^10, p=(k<<(n+1))+1; if(isprime(p), break())); e=p*(p-1)/(1<<(n+1)); h=znprimroot(p^2)^e; g=h^2; m=p^2; for(i=1,1<<n,m=min(m,lift(h));h*=g); m

%Y Cf. A260824, A248214, A035089.

%K nonn

%O 0,1

%A _Jeppe Stig Nielsen_, Aug 08 2015

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified December 1 22:12 EST 2021. Contains 349435 sequences. (Running on oeis4.)