This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
A086892 := proc(n) gcd(2^n-1,3^n-1) ; end: for n from 1 to 40 do printf("%d,",A086892(11*n)) ; od: # R. J. Mathar, Aug 12 2008
Since 2^5-1 = 31 and 3^5-1 = 242 are relatively prime, a(5) = 3. The divisors of 72 are 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72; Adding one to each divisor gives 3, 4, 5, 7, 9, 10, 13, 19, 25, 37, 73; The odd primes in this list are 3, 5, 7, 13, 19, 37, 73; The product of these primes is 3 * 5 * 7 * 13 * 19 * 37 * 73 = 70050435; Thus a(72) is of the form 70050435 * m.
f:= proc(n) local P, Q, M, k, kn;
P:= select(isprime, map(t -> t+1, numtheory:-divisors(n)) minus {2});
M:= convert(P,`*`);
Q:= 2^n - 1;
for k from M by 2*M do
kn:= k &^ n - 1 mod Q;
if igcd(kn, Q) = 1 then
return k
fi
od
end proc:
map(f, [$1..100]); # Robert Israel, Sep 02 2015
Table[k = 1; While[! CoprimeQ[2^n - 1, k^n - 1], k++]; k, {n, 59}] (* Michael De Vlieger, Sep 01 2015 *)
a(n) = {my(k=1, pt = 2^n-1); while (gcd(pt, k^n-1) != 1, k++); k;} \\ Michel Marcus, Jul 17 2015 && Jan 27 2016
conjecture_a(n) = {my(d=divisors(n)); v=select(x->isprime(x+1),select(y->y%2==0,d)); v+= vector(#v,i,1); p=prod(i=1,#v,v[i]); if(p==1&&n!=1,p=3);
forstep(i=1, p, 2,if(gcd((p * i)^n-1, 2^n-1)==1, return(p*i))); for(i=1,n,if(gcd(p^n-1,2^n-1) == 1, return(p), p=nextprime(p+1))); forstep(i=1,100000,2,if(gcd((i)^n-1,2^n-1)==1,return(i)))} \\ David A. Corneth, Sep 01 2015
def min_k(n):
g,k=2,0
while g!=1:
k=k+1
g=gcd(2^n-1,k^n-1)
return k
print([min_k(n) for n in [1..71]])
For n=3, 2^3-1 = 7 and 5^3-1 = 124, thus a(3) = gcd(7,124) = 1.
seq(igcd(2^n-1, 5^n-1), n=1..100);
Table[GCD[2^n - 1, 5^n - 1], {n, 100}]
vector(100,n,gcd(2^n-1,5^n-1))
[gcd(2^n-1,5^n-1) for n in [1..100]]
A010696 := proc(n) op( (n mod 2)+1,[2,6]) ; end: A086892 := proc(n) gcd(2^n-1,3^n-1) ; end: A141498 := proc(n) A010696(n-1)* A086892(n) ; end: A027760 := proc(n) local s, p; s := 0 ; p := 2; while p <= n+1 do if n mod (p-1) = 0 then s := s+1/p; fi; p := nextprime(p) ; od: denom(s) ; end: A141517 := proc(n) A141498(n)/A027760(n) ; end: seq(A141517(n),n=1..120) ; # R. J. Mathar, Sep 03 2009
Since 3^5-1 = 242 and 2^5-1 = 31 are relatively prime, a(5) = 2.
f:= proc(n) local t,F,m,k,v;
t:= 3^n-1;
F:= select(isprime,map(`+`,numtheory:-divisors(n),1));
m:= convert(select(s -> t mod s = 0, F),`*`);
for k from m by m do
v:= k &^ n - 1 mod t;
if igcd(v, t) = 1 then return k fi
od
end proc:
map(f, [$1..100]); # Robert Israel, Nov 20 2024
Table[k = 1; While[! CoprimeQ[3^n - 1, k^n - 1], k++]; k, {n, 59}] (* Michael De Vlieger, Jan 27 2016 *)
a(n) = {k=1; while( gcd(3^n-1, k^n-1)!=1, k++); k; }
def min_k(n):
g, k=2, 0
while g!=1:
k=k+1
g=gcd(3^n-1, k^n-1)
return k
print([min_k(n) for n in [1..60]])
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