login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A190530 Numbers k such that 2^k - 3 is not squarefree. 0

%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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified March 28 08:22 EDT 2024. Contains 371236 sequences. (Running on oeis4.)