A190530 Numbers k such that 2^k - 3 is not squarefree.

7, 16, 27, 47, 67, 87, 88, 107, 124, 127, 139, 147, 162, 167, 187, 198, 207, 227, 247, 267, 272, 280, 287, 303, 307, 308, 314, 327, 341, 347, 367, 387, 407, 415, 418, 427, 436, 447, 467, 481, 485, 487, 507, 514, 527, 528, 537, 547, 567, 587, 592, 607, 627, 638, 647

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

Links

Table of n, a(n) for n=1..55.

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

Sequence in context: A119461 A326664 A028560 * A345071 A351044 A133694

Adjacent sequences: A190527 A190528 A190529 * A190531 A190532 A190533

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