|
%I #21 Nov 23 2023 09:11:39
%S 1,2,3,5,7,9,13,17,19,31,61,67,89,107,127,137
%N Numbers k such that 2*(2^k-1) is in A354525.
%C Numbers k such that for every prime factor p of 2^k-1 we have gpf(2*(2^k-1)+p) = p.
%C Numbers k such that for every prime factor p of 2^k-1, 2*(2^k-1)+p is p-smooth.
%C All terms except 2 are odd: if k is even, then 3 is a factor of 2^k-1, so 3^m = 2*(2^k-1)+3 = 2^(k+1) + 1 => k+1 >= 3^(m-1). The only possible case is (k,m) = (2,2).
%C Clearly A000043 is a subsequence. The exceptional terms (1, 9, 67, 137, ...) are listed in A354532.
%C The next term is >= 349. The next composite term, if it exists, is >= 7921 = 89^2.
%e See A354532.
%o (PARI) gpf(n) = vecmax(factor(n)[, 1]);
%o ispsmooth(n,p,{lim=1<<256}) = if(n<=lim, n==1 || gpf(n)<=p, my(N=n/p^valuation(n,p)); forprime(q=2, p, N=N/q^valuation(N,q); if((N<=lim && isprime(N)) || N==1, return(N<=p))); 0); \\ check if n is p-smooth, using brute force if n is too large
%o isA354531(n,{lim=256},{p_lim=1<<32}) = {
%o my(N=2^n-1);
%o if(isprime(N), return(1));
%o if(n>lim, forprime(p=3, p_lim, if(N%p==0 && !ispsmooth(2*N+p,p), return(0)))); \\ first check if there is a prime factor p <= p_lim of 2^n-1 such that 2*(2^n-1)+p is not p-smooth (for large n)
%o my(d=divisors(n));
%o for(i=1, #d, my(f=factor(2^d[i]-1)[, 1]); for(j=1, #f, if(!ispsmooth(2*N+f[j],f[j],1<<lim), return(0)))); 1 \\ then check if 2*(2^n-1)+p is p-smooth for p|2^d-1, d|N
%o }
%Y Cf. A354525, A000043, A354532, A354533, A354536.
%K nonn,hard,more
%O 1,2
%A _Jianing Song_, Aug 16 2022
|
