|
|
A186253
|
|
Indices of zeros of the sequence u(n)=abs(u(n-1)-gcd(u(n-1),n-1)), u(1)=1.
|
|
21
|
|
|
2, 5, 11, 23, 47, 79, 157, 313, 619, 1237, 2473, 4909, 9817, 19603, 39199, 78193, 156019, 311347, 622669, 1244149, 2487739, 4975111, 9950221, 19900399, 39800797, 79601461, 159202369, 318404629, 636788881, 1273577761, 2547155419, 5094310069, 10188620041
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,1
|
|
COMMENTS
|
For any fixed integer m>=1 define u(1)=1 and u(n)=abs(u(n-1)-gcd(u(n-1),m*n-1)). Then (b_m(k))_{k>=1} is the sequence of integers such that u(b_m(k))=0 and we conjecture that for k large enough m*b_m(k)+m-1 is a prime number. Here for m=1 it appears a(n) is prime for n>=1.
|
|
LINKS
|
|
|
FORMULA
|
Conjecture: a(n) is asymptotic to c*2^n with c = 1.1861...
|
|
MATHEMATICA
|
a = m = 1; Reap[For[n = 2, n <= 10^7, n++, a = Abs[a - GCD[a, m*n - 1]]; If[a == 0, Print[m*n + m - 1]; Sow[m*n + m - 1]]]][[2, 1]] (* Jean-François Alcover, Feb 05 2019, from PARI *)
nxt[{n_, a_}]:={n+1, Abs[a-GCD[a, n]]}; Position[NestList[nxt, {1, 1}, 13*10^5][[All, 2]], 0]// Flatten (* The program generates the first 20 terms of the sequence. *) (* Harvey P. Dale, Oct 02 2022 *)
|
|
PROG
|
(PARI) a=1; m=1; for(n=2, 1e7, a=abs(a-gcd(a, m*n-1)); if(a==0, print1(m*n+m-1, ", ")))
(PARI)
next_a(last_a) = {
local(A=last_a, B=last_a, C=2*last_a+1);
while(A>0,
D=divisors(C);
k1=10*D[2];
for(j=2, #D, d=D[j]; k=((A+1-B+d)/2)%d;
if(k==0, k=d); if(k<=k1, k1=k; d1=d));
if(k1-1+d1==A, B=B+1);
A = max(A-(k1-1)-d1, 0);
B = B + k1;
C = C - (d1 - 1);
);
return(B);
}
a=2
(PARI) m=a=k=1; for(n=1, 30, while( a>d=vecmin(apply(p->a%p, factor(N=m*(k+a)+m-1)[, 1])), a-=d+gcd(a-d, N); k+=1+d); k+=a+1; print1(a=N, ", ")) \\ M. F. Hasler, Aug 22 2015
(Haskell)
a186253 n = a186253_list !! (n-1)
a186253_list = filter ((== 0) . a261301) [1..]
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|