A217448 Least k > 0 such that 1 + n^2 and 1 + (n+k)^2 have the same smallest prime factor.

2, 6, 2, 26, 2, 74, 2, 4, 2, 404, 2, 6, 2, 366, 2, 514, 2, 4, 2, 1564, 2, 6, 2, 1106, 2, 4010, 2, 4, 2, 34, 2, 6, 2, 10, 2, 2594, 2, 4, 2, 22334, 2, 6, 2, 16, 2, 58, 2, 4, 2, 64, 2, 6, 2, 29062, 2, 18710, 2, 4, 2, 10, 2, 6, 2, 42, 2, 17428, 2, 4, 2, 16, 2, 6

Offset

1,1

Comments

Alternate title: Least k > 0 such that A089120(n) = A089120(n+k).

A089120(n): smallest prime factor of n^2 + 1.

Conjecture: a(n) exists for all n.

Links

Antti Karttunen, Table of n, a(n) for n = 1..10000

Example

a(10) = 404 because 10^2 + 1 = 101, (10+404)^2+1 = 101*1697 so A089120(10) = A089120(414) = 101;
a(170) = 404274 because 170^2 + 1 = 28901, (170+404274)^2+1 = 163574949137 = 28901* 5659837 so A089120(170) = A089120(40444) = 28901.

Maple

with(numtheory):T:=array(1..100): for n from 1 to 100 do:x:=factorset(n^2+1):n1:=nops(x): T[n] := x[1]:od:for a from 1 to 80 do:p:=T[a]:ii:=0:for k from 1 to 50000 while(ii=0) do: z:=factorset((a+k)^2+1): n2:=nops(z):if z[1]=p then printf(`%d, `, k):ii:=1:else fi:od:od:

Mathematica

sspf[n_]:=Module[{c=FactorInteger[1+n^2][[1, 1]], k=1}, While[ FactorInteger[ 1+ (n+k)^2][[1, 1]]!=c, k++]; k]; Array[sspf, 80] (* Harvey P. Dale, Oct 12 2012 *)

Prog

(PARI)
A020639(n) = if(1==n, n, factor(n)[1, 1]);
A217448(n) = { my(spf=A020639(1+(n^2)), x); for(k=1, oo, x=1+((n+k)^2); if(!(x%spf) && A020639(x)==spf, return(k))); }; \\ Antti Karttunen, May 24 2021

Crossrefs

Cf. A089120, A217393, A014442.

Sequence in context: A125032 A076743 A131980 * A280705 A027760 A141056

Adjacent sequences: A217445 A217446 A217447 * A217449 A217450 A217451

Keyword

nonn,look

Author

Michel Lagneau, Oct 03 2012

Status

approved