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A237043 Numbers n such that 2^n - 1 is not squarefree, but 2^d - 1 is squarefree for every proper divisor d of n. 12
6, 20, 21, 110, 136, 155, 253, 364, 602, 657, 812, 889, 979, 1081 (list; graph; refs; listen; history; text; internal format)



Primitive elements of A049094: the elements of A049094 are precisely the positive multiples of members of this sequence.

If p^2 divides 2^n - 1 for some odd prime p, then by definition the multiplicative order of 2 mod p^2 divides n. The multiplicative order of 2 mod p^2 is p times the multiplicative order of 2 mod p unless p is a Wieferich prime, in which case the two orders are identical. Hence either p is a Wieferich prime or p*log_2(p+1) <= n. This should allow finding larger members of this sequence. - Charles R Greathouse IV, Feb 04 2014

If n is in the sequence and m>1 then m*n is not in the sequence. Because n is a proper divisor of m*n and 2^n-1 is not squarefree. - Farideh Firoozbakht, Feb 11 2014

a(15)>=1207. - Max Alekseyev, Sep 28 2015


Table of n, a(n) for n=1..14.


(PARI) default(factor_add_primes, 1);

isA049094(n)=my(f=factor(n>>valuation(n, 2))[, 1], N, o); for(i=1, #f, if(n%(f[i]-1) == 0, return(1))); N=2^n-1; fordiv(n, d, f=factor(2^d-1)[, 1]; for(i=1, #f, if(d==n, return(!issquarefree(N))); o=valuation(N, f[i]); if(o>1, return(1)); N/=f[i]^o))

is(n)=fordiv(n, d, if(isA049094(d), return(d==n))); 0

(PARI) \\ Simpler but slow

is(n)=fordiv(n, d, if(!issquarefree(2^d-1), return(d==n))); 0


Cf. A049094, A005420, A065069, A282631, A282632.

Sequence in context: A222604 A112809 A289659 * A243905 A062017 A103678

Adjacent sequences:  A237040 A237041 A237042 * A237044 A237045 A237046




Charles R Greathouse IV, Feb 02 2014


a(14) from Charles R Greathouse IV, Sep 21 2015, following Womack's factorization of 2^991-1.



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Last modified January 20 02:45 EST 2018. Contains 297939 sequences.