OFFSET
1,1
COMMENTS
If for some prime p such that 2^p-1 is also prime one has n = 2^(p+1) - 3 an odd perfect number, (that is in particular not squarefree), then the even perfect number m = 2^(p-1)(2^p-1) and the odd perfect number n, are the second pair of perfect numbers (m,n) with m > n such that m/(n+1) is a power of 2. Indeed we have in this case: m/(n+1)=2^(p-2). The only other known pair of such numbers is (m,n) = (28,6), so that m/(n+1) = 28/7 = 2^2.
Other terms in this sequence: 647, 667, 668, 687, 707, 727, 747, 748, 767, 787, 807, 823, 827, 847, 858, 867, 887, 904, 907, 921, 927, 947, 967, 968, 987. - Kevin P. Thompson, May 22 2022
Contains all numbers k of the forms 20*j + 7 and 110*j + 88 (where 2^k - 3 is divisible by 5^2 and 11^2 respectively). - Robert Israel, May 22 2023
EXAMPLE
a(7) = 88, since 2^88 - 3 is divisible by 11^2, so that it is not squarefree.
MATHEMATICA
Select[Range[100], !SquareFreeQ[2^# - 3] &] (* Amiram Eldar, Dec 24 2020 *)
PROG
(PARI) for(n=1, 1000, if(!issquarefree(2^n-3), print1(n, ", "))) /* Joerg Arndt, May 13 2011 */
CROSSREFS
KEYWORD
nonn
AUTHOR
Luis H. Gallardo, May 11 2011
EXTENSIONS
a(20)-a(44) from Charles R Greathouse IV, May 11 2011
a(45)-a(51) from Amiram Eldar, Dec 24 2020
a(52)-a(54) from Kevin P. Thompson, May 22 2022
a(55) from Kevin P. Thompson, Sep 04 2022
STATUS
approved