%I #31 May 23 2023 06:03:10
%S 7,16,27,47,67,87,88,107,124,127,139,147,162,167,187,198,207,227,247,
%T 267,272,280,287,303,307,308,314,327,341,347,367,387,407,415,418,427,
%U 436,447,467,481,485,487,507,514,527,528,537,547,567,587,592,607,627,638,647
%N Numbers k such that 2^k - 3 is not squarefree.
%C 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.
%C 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
%C 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
%e a(7) = 88, since 2^88 - 3 is divisible by 11^2, so that it is not squarefree.
%t Select[Range[100], !SquareFreeQ[2^# - 3] &] (* _Amiram Eldar_, Dec 24 2020 *)
%o (PARI) for(n=1,1000,if(!issquarefree(2^n-3),print1(n,", "))) /* _Joerg Arndt_, May 13 2011 */
%K nonn
%O 1,1
%A _Luis H. Gallardo_, May 11 2011
%E a(20)-a(44) from _Charles R Greathouse IV_, May 11 2011
%E a(45)-a(51) from _Amiram Eldar_, Dec 24 2020
%E a(52)-a(54) from _Kevin P. Thompson_, May 22 2022
%E a(55) from _Kevin P. Thompson_, Sep 04 2022
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