login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A274721 a(n) is the least k such that A051903(k^2+1) = n. 1
1, 7, 57, 182, 2057, 1068, 32318, 110443, 280182, 3626068, 23157318, 120813568, 123327057, 1097376068, 11109655182, 49925501068, 407838170807, 355101282318, 3459595983307, 15613890344818, 365855836217682, 110981321985443, 2273204469030182, 9647724486047943 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Least k such that the largest exponent of a prime in the factorization of k^2+1 is n.

Conjecture: for each n > 1, a(n) = A034939(n) or 5^n - A034939(n).

For any n > 1, -1 has two square roots mod 5^n; at least one of these is not a square root of -1 mod 5^(n+1).  If v is this number, v < 5^n so v^2 < 25^n.  v^2+1 might be divisible by p^(n+1) for p = 13 or 17, or a square root of -1 mod 13^n or 17^n might be smaller than v, but that seems very unlikely.  Thus the conjecture.

LINKS

Robert Israel, Table of n, a(n) for n = 1..109

EXAMPLE

1^2 + 1 = 2.

7^2 + 1 = 2*5^2.

57^2 + 1 = 2*5^3*13.

182^2 + 1 = 5^4 * 53.

MAPLE

F:= proc(n) local v, p, w;

  v:= numtheory:-msqrt(-1, 5^n);

v:= min(v, 5^n-v);

if max(seq(t[2], t=ifactors(v^2+1)[2])) > n then

    v:= 5^n - v;

    if max(seq(t[2], t=ifactors(v^2+1)[2])) > n then

         error "neither %d nor %d works", 5^n-v, v fi

fi;

for p from 13 by 4 while p^n <= v^2+1 do

    if isprime(p) then

     w:= numtheory:-msqrt(-1, p^n);

     w:= min(w, p^n-w);

     if w < v then

        if max(seq(t[2], t=ifactors(w^2+1)[2])) = n then

           v:= w;

        fi

     fi

    fi

od;

v

end proc:

F(1):= 1:

map(F, [$1..100]);

CROSSREFS

Cf. A034939, A051903.

Sequence in context: A110830 A304691 A218428 * A180817 A201438 A202510

Adjacent sequences:  A274718 A274719 A274720 * A274722 A274723 A274724

KEYWORD

nonn

AUTHOR

Robert Israel, Jul 14 2016

STATUS

approved

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.

License Agreements, Terms of Use, Privacy Policy. .

Last modified October 18 14:11 EDT 2021. Contains 348068 sequences. (Running on oeis4.)