A260872 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.

1, 33, 57, 73, 105, 129, 145, 177, 193, 201, 217, 249, 273, 313, 337, 345, 385, 393, 409, 457, 465, 481, 537, 553, 561, 577, 609, 633, 649, 673, 697, 705, 745, 753, 777, 793, 817, 849, 865, 889, 897, 913, 921, 969, 985, 1009, 1041, 1065, 1081, 1113, 1129

Offset: 1

Author

J. Lowell, Aug 01 2015

Keywords

• nonn

Comments

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.

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

Examples

			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.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

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

Programs

Mathematica

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