A260872 - OEIS

%I #25 Jun 09 2022 19:37:33

%S 1,33,57,73,105,129,145,177,193,201,217,249,273,313,337,345,385,393,

%T 409,457,465,481,537,553,561,577,609,633,649,673,697,705,745,753,777,

%U 793,817,849,865,889,897,913,921,969,985,1009,1041,1065,1081,1113,1129

%N Squarefree numbers k such that k+1 has no primes of the form 4*m-1 and at most one 2 in its prime factorization.

%C An even number k is congruent to either 0 or 2 mod 4. If congruent to 0, it is divisible by 4 and thus not squarefree. If k is congruent to 2, k+1 will be one less than a multiple of 4, and thus at least one prime factor of k+1 will be one less than a multiple of 4. Thus, there are no even numbers in this sequence.

%C From the author's comment above, all sequence terms must be odd, so k+1 must always be even and k+1 will always be singly even. - _Ray Chandler_, Aug 03 2015

%H Amiram Eldar, <a href="/A260872/b260872.txt">Table of n, a(n) for n = 1..10000</a>

%e 41 + 1 = 42 = 2*3*7 and both 3 and 7 are prime numbers of the form 4*n-1, so 41 is not a term of this sequence.

%t Select[Range[1100],SquareFreeQ[#]&&IntegerExponent[#+1,2]<2&&Select[First/@FactorInteger[#+1],Mod[#,4]==3&]=={}&] (* _Ray Chandler_, Aug 02 2015 *)

%Y Cf. A002144, A002145, A004613, A004614, A005117, A016825.

%K nonn

%O 1,2

%A _J. Lowell_, Aug 01 2015