A381502 - OEIS

A381502

a(n) is the smallest number k such that with x(1) = k and x(i+1) = 2*x(i) + 1, x(i+1) has exactly one more prime factor (counted with multiplicity) than x(i) for i = 1 to n but not i = n + 1.

2, 1, 87, 43, 197161, 8651161

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OFFSET

1,1

COMMENTS

a(n) is the least number m such that A001222(2^k * (m+1) - 1) = A001222(m) + k for 1 <= k <= n but not for k = n + 1.

a(7) > 10^9 if it exists.

LINKS

Table of n, a(n) for n=1..6.

FORMULA

A001222(2^k * (a(n) + 1) - 1) = A001222(a(n)) + k for k <= n but not for k = n + 1.

EXAMPLE

a(4) = 43 because 43 is prime, 2*43 + 1 = 87 = 3 * 29 has 2 prime factors, 2*87 + 1 = 175 = 5^2*7 has 3, and 2*175 + 1 = 351 = 3^3 * 13 has 4, but 2*351 + 1 = 703 = 19 * 37 has only 2.

MAPLE

f:= proc(n) local x, m, i;

m:= numtheory:-bigomega(n); x:= n;

for i from 1 do

x:= 2*x+1;

if numtheory:-bigomega(x) <> m+i then return i fi

end proc:

N:= 6: V:= Vector(N): count:= 0:

for i from 1 while count < N do

r:= f(i);

if r <= N and V[r] = 0 then V[r]:= i; count:= count+1 fi;

od: